How do you find the derivative of #(cos x)^2 - cos x#? How do you differentiate #f(x)=cos(sqrt(3+e^(x^2)))# using the chain rule? How do you differentiate #f(x)=csc(sqrt(x^2-5x)) # using the chain rule? How do you find #f^37x# given #f(x)=cos3x#? How do you differentiate #f(x)=csc(1/x^3) # using the chain rule? How do you differentiate #f(x)=tan(e^(1/x)) # using the chain rule? What is the derivative of #w =sqrt(x^2+y^2+z^2)#? How do you differentiate #f(x)=sqrt(x-(3x+5)^2)# using the chain rule.? How do you differentiate #arcsin(csc(1-1/x^3)) )# using the chain rule? If #f(x)= sin3x # and #g(x) = 2x^2 -3x #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=e^(x^2-3x^2-4) # using the chain rule? If #f(x)= x^2-x # and #g(x) = x^( 1/3 ) #, what is #f'(g(x)) #? The chain rule is often one of the hardest concepts for calculus students to understand. The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. How do you find the derivative of #h(x)= 1/(6x^2+x+1)^2#? How do you differentiate #y=sqrt(2-e^x)#? How do you find the derivative of #x*(sqrt(4-x^2))#? How do you differentiate # y= cos(pi/2x^2-pix) # using the chain rule? Click HERE to return to the list of problems. How do you find the derivative of #y= [(2x+3)/(x-2)][(5x-1)/(3x-2)]# using the chain rule? How do you differentiate #tan(cos^3(x))#? How do you find the derivative of #f(x) = -15 / (4x + 5)^4# using the chain rule? which represents the slope of the tangent line at the point (−1,−32). Chain Rule The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. So when using the chain rule: How do you differentiate #f(x)=sec^4(e^(x^3) ) # using the chain rule? Let’s use the second form of the Chain rule above: If #f(x)= 2 x^2 - 3 x # and #g(x) = 2e^x + 1 #, how do you differentiate #f(g(x)) # using the chain rule? If #f(x)= 2 x^2 + x # and #g(x) = sqrtx + 1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate # f(x)=tan(e^((lnx-2)^2 ))# using the chain rule.? How do you differentiate #f(x)=x sec kx^2 # using the chain rule? How do you find the derivative of # y= sqrt(x^2 + cos x)# using the chain rule? Is there a chain rule for partial derivatives? How do you find the derivative of # f(x) = sin ^{ 3 }x# using the chain rule? How do you calculate the derivative for #y = 3(5 - x^2)^5#? How do you find the derivative of # y = ln(secx)#? WHEN TO USE CHAIN RULE. How do you find the derivative of #ln(sqrtx)#? How do you use the chain rule to differentiate #y=1/(x^4+x)^2#? How do you use the chain rule to differentiate #ln(4x)^10#? How do you differentiate # y = 1/4 [ 1/2ln[x^2 -2x +2/ x^2 + 2x +2] + tan^-1[2x/2-x^2]]#? How do you differentiate # y =x /sec ^2x^3# using the chain rule? How do you find the derivative of #(1-y^2)^(1/2)#? Because the slope of the tangent line to a curve is the derivative, you find that. How do you find f'(x) for #f(x) = (ln x)^8#? How do I find the derivative of #y= arccos (e^7x)#? We know: We just have to find our two functions, find their derivatives and input into the Chain Rule expression. Now, for the first of these we need to apply the product rule first: To find the derivative inside the parenthesis we need to apply the chain rule. If #f(x) =-e^(x) # and #g(x) = 3csc^2x^2 #, what is #f'(g(x)) #? How do you differentiate # f(x)=e^((lnx-2)^2 # using the chain rule.? How do you find the derivative of #u=(6-2x^2)^3#? How do you differentiate #f(x)=cos(x^3)#? This is for both equations. How do you differentiate #f(x)=1/sin(e^arctanx)# using the chain rule? If #f(x)= 3x-2 # and #g(x) = e^x #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=(x^2+5x-2)^2#? What is the derivative of #ln sqrt(2x+1)#? How do you calculate the derivative of #r= 2thetasqrt(sec theta)#? How do you find the derivative of #sqrt (x^2 +1)#? The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.An example of one of these types of functions is $$f(x) = (1 + x)^2$$ which is formed by taking the function … How do you differentiate #y=e^((lnx))^2#? What is the derivative of? How do you differentiate #f(x)=ln(1/sin(3x))# using the chain rule? How do you use the chain rule to differentiate #y=5/(2x^3+3x)#? What is the derivative of #sqrt(e^(2x) +e^(-2x))#? How do you differentiate #f(x)=4(x^2 + x - 1)^10 # using the chain rule? If #f(x)= - e^(5x # and #g(x) = 2x^3 #, how do you differentiate #f(g(x)) # using the chain rule? Differentiation of Inverse Trigonometric Functions, Differentiation of Exponential and Logarithmic Functions, Volumes of Solids with Known Cross Sections. How do you use the chain rule to differentiate #f(x) = e^(4x+9)#? How do you differentiate #f(x)=(4x-x^2)^100#? How do you differentiate #y = ln [x^4 sin^2 (x)]#? If #f(x) =xe^x# and #g(x) = e^(3x)#, what is #f'(g(x)) #? What is the derivative of #ln(x^2)+5^(2x)#? How do you use the chain rule to differentiate #y=(x^2+4)^-1#? How do you differentiate #tan(3x^2) - csc ( ln(4x) )^2#? How do you differentiate #f(x) = (x^3+cos3x)^(1/2)?# using the chain rule? Now, let's differentiate the same equation using the chain rule which states that the derivative of a composite function equals: (derivative of outside) • (inside) • (derivative of inside). If #f(x) =-sqrt(3x-1) # and #g(x) = (x+3)^3 #, what is #f'(g(x)) #? How do you find the derivative of #y= sin(sin(sin(x)))# ? How do you use the chain rule to differentiate #y=(3x^4-7x^3+3x^2-5x)^3#? How do you find the derivative of #g(x)=3(2-5x)^6#? What is the derivative of #(cos x)^(sin x)#? What is the derivative of #sin(x + (π/2)) #? WHEN TO USE CHAIN RULE. Since the functions were linear, this example was trivial. When do you use the chain rule instead of the product rule? How do you use the chain rule to differentiate #y=x^3(2x-5)^4#? What is the derivative of # ( cos (pi*x) +1 ) / x#? Solution for dw Use chain rule to find dt if w = xez, x = t², y = 1- t, z = 1+ 2t. How do you differentiate # y = (6e^(-7x)+2x)^2# using the chain rule? Example 60: Using the Chain Rule. How do you differentiate # f(x)=sqrt(ln(xe^x))# using the chain rule.? If #f(x)= sqrt(x-2 # and #g(x) = e^(2x #, what is #f'(g(x)) #? Chain rule is also often used with quotient rule. What is the derivatives of #sec2x# and #tan2x#? How do you find the derivative of # (y - 1)^5 / (y^2 + 5 y)^6# using the chain rule? How do you differentiate #y=2^(3^(x^2))#? How do you differentiate #f(x)=sqrt(3+x^2) # using the chain rule? How do you differentiate #e^(2x^2+x) # using the chain rule? Step 1. How do you use the chain rule to differentiate #y=(x+1)^(1/2)#? What is the derivative of #sin^2(x) + sinx#? How do you find the derivative of # pi^(x+2)# using the chain rule? How do you differentiate #f(x)=cot(sqrt(x-3)) # using the chain rule? How do you differentiate #f(x)=sin(e^(3-x)) # using the chain rule? How do you find the derivative of #root3(x^-5)#? What is the derivative of #f(x) = x(sqrt( 1 - x^2))#? What is the first differential of #y = t^(3/2)(16-sqrtt)#? How do you differentiate #f(x) = xcos((pix)/2)# using the chain rule? How do you differentiate #f(x)=e^(secsqrtx)# using the chain rule.? How do you differentiate #y=3cot(ntheta)#? What is the derivative of #(ln x)^(1/5)#? How do you find the derivative for #1/sqrt(1-x^2)#? How do you differentiate #f(x)=sqrt(csc(1/x^3 ) # using the chain rule? One way to do that is through some trigonometric identities. How do you differentiate #f(x) = (3x-2)^4# using the chain rule? If #f(x) =-sqrt(2x-1) # and #g(x) = 3/x^3 #, what is #f'(g(x)) #? How do you differentiate #f(x)=ln(sin(e^{x}))#? How do you find the derivative of #f(x)=(8x+3)^.5#? However, we rarely use this formal approach when applying the chain rule to … How do you find the derivative of #sqrt(x ln(x^4))#? How do you find the derivative of #f(x) = cos(pi/2)x# using the chain rule? How do you differentiate #f(x)=sqrt(cos^3(3x-2))# using the chain rule? How do you differentiate # f(x)=ln(6x+8)# using the chain rule.? Just use the rule for the derivative of sine, not touching the inside stuff (x 2), and then multiply your result by the derivative of x 2. How do you find the derivative of #y=ln(sin2x)#? How do you differentiate #f(x) = sin(sqrt(arccosx^2)) # using the chain rule? How do you differentiate #y = (sqrt(cos x))/(5lnx)#? How do you differentiate #f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9# ? How do you find the derivative of #y=x*sqrt(16-x^2)#? How do you find #(d^2y)/(dx^2)# given #2x^2-3y^2=4#? If #f(x)= - e^x # and #g(x) = sqrt(1-x #, how do you differentiate #f(g(x)) # using the chain rule? How do you find #f'(0)# if #f(x)=sin^2(3-x)#? What is the derivative of #x^3 * (2/3x^2 -1)^4#? What's the derivative of #f(x)=g(x)^(h(x))#? How do you use the chain rule to differentiate #sqrt(4x+9)#? if #v=1148sqrt(p)# where #p# is a function of #t# then find #(dv)/dt# when #p=44.0# and #dp/dt=0.307#? What is the derivative of #(sqrtx-1)/sqrtx#? What is the derivative of #y = sin(tan(5x))#? How do you find the derivative of #f(x)=1/(2x+5)^5#? What is the derivative of #y= (5x)/sqrt (x^2+9)#? How do you differentiate #f(x)=sec(1/x^3)# using the chain rule? How do you differentiate given # 12(sin5x)^3#? How do you find the derivative of #f(x)=(6-5x)^-1#? How do you find the first and second derivative of #h(x)=sqrt(x^2+1)#? The chain rule is used when you want to differentiate a function to the power of a number. How do you differentiate #f(x)=e^(csc2x)# using the chain rule.? It is useful when finding the derivative of a function that is raised to the nth power. How do you find the derivative of #e^ [2 tan(sqrt x)]#? How do you differentiate #f(x)=tansqrtx# using the chain rule? How do you differentiate #arcsin(csc(4x)) )# using the chain rule? How do you use the chain rule to differentiate #y=((x^5+4)/(x^2-5))^(1/5)#? How do you differentiate # f(x)=sqrt(ln(1/(xe^x))# using the chain rule.? How do you find the second derivative of #f(x)=ln(7x^2e^x\sin x)#? How do you find the derivative of # y= ln (1 - x^2)#? The chain rule is a rule for differentiating compositions of functions. How do you differentiate #f(x)=sqrt(tane^(4x)# using the chain rule.? How do you differentiate #f(x)=-5 xe^(x/cos x)# using the chain rule? If #f(x)= csc7 x # and #g(x) = e^(1 +3x ) #, how do you differentiate #f(g(x)) # using the chain rule? If #f(x)= tan2 x # and #g(x) = sqrt(-4x-3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate # f(x)=e^(x-(x-2)^2 # using the chain rule.? How do you differentiate given #sec^2(x) #? How do you differentiate #f(x)=sqrt(((3x)/(2x-3))# using the chain rule? How do you find the derivative of #18lnx+x^2+5#? Before we discuss the Chain Rule formula, let us give another example. How do you find the derivative for #g(x)= 3tan4x#? How do you differentiate #g(x)=(1+4x)^5(3+x-x^2)^8#? This is the final formula that we use in backpropagation. How do you find the derivative of #y = 3 / (cos2x^2)#? How do you differentiate #y = e^(-2x + x^2)#? Indeed, we have So we will use the product formula to get And I'll have a special version of the chain rule that I'll use for these and I'll call this rule the general exponential rule. It states: The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. How do you find the derivative of # f(t) = sin(e^t) + e^(sint)#? How do you differentiate #f(x)=e^(cotsqrtx)# using the chain rule.? What is the derivative of #(1+4x)^5(3+x-x^2)^8#? How do you differentiate #f(x) = e^(e^x)#? What is the derivative of #y = ln(2x^3-x^2)#? How do you use the chain rule to differentiate #y=-5/(3x^2-4)^6#? How do you differentiate #f(x)=sec(1/sqrt(3x^2-4) ) # using the chain rule? How do you find the derivative of #(1+x)^(1/x)#? For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. How do you find the derivative of #3(x^2-2)^4#? How do you find the derivative of the function #f(w) = ln(sin(w−15))#? How do you find the derivative of #r(x)= (0.3x-4.9x^-1)^0.5#? If #f(x) = -x -2# and #g(x) = e^(x^2-x)#, what is #f'(g(x)) #? Click HERE to return to the list of problems. #f'(x)=?# How do you find the derivative of #y=ln((x+sqrt(1+x^2))/(1+sqrt2))#? What is the derivative of #g(x)=sqrt(5-3x)#? Most problems are average. How do you differentiate #y=(x^4+3x^2-2)^5#? What is the derivative of # ln[-30(x^3-2x+e^x)^5]#? If #F(x)=f(xf(xf(x)))# where f(1)=2, f(2)=3, f'(1)=4, f'(2)=5, and f'(3)=6, how do you find F'(1)? How do you differentiate # f(x)=e^((6x-2)^2 # using the chain rule.? How do you differentiate #f(x) = (1-3sqrt(2x^2-1))^2 # using the chain rule? How do you find the derivative of #y=-(2x+3+4x^-1)^-1#? Find the first and second derivative of #(2lnx)/x#? How do you differentiate # y =-sqrt(e^(x-sin^2x)# using the chain rule? The chain rule applies whenever you have a function of a function or expression. How do you differentiate # f(x)=sin(e^((lnx-2)^2 ))# using the chain rule.? What is the derivative of #(sqrt 6)/x^5# using the Power Rule? How do you find the derivative of #y=ln(e^x+3)# ? How do you use the chain rule to find the derivative of log x? When to Use Chain Rule. How do you find the derivative of #sqrt(1-x^2)#? How do you differentiate #f(x) = sqrt((3x)/(2x-3))# using the chain rule? How do you use the chain rule to differentiate #y=4/(sqrt(x-5)#? How do you differentiate #f(x)=ln(sine^(x^2))# using the chain rule? Chain Rule #=>y'=3((1+x)/(1-x))^2*((1+x)/(1-x))'#, #y'=3((1+x)/(1-x))^2*((1-x)(1)-(1+x)(-1))/(1-x)^2#, #y'=3((1+x)/(1-x))^2*((1-x)-(1+x)(-1))/(1-x)^2#, #y'=3((1+x)/(1-x))^2*((1-x)+(1+x))/(1-x)^2#, #y'=1/4(x^2+3x+5)^{1/4-1}cdot(x^2+3x+5)'#. How do you differentiate #y = (x+1) (sqrt (2x-1))#? This calculator … How do you differentiate # f(x)=e^((ln(x^2+3)^2)# using the chain rule.? How do you differentiate # y = 17(22+x)^((41-x)^30)#? If #f(x) =xe^x# and #g(x) = sinx-x#, what is #f'(g(x)) #? How do you find the derivative of #lnsqrt x#? How do you use the chain rule to differentiate #y=1/(x^4-1)#? How do you find the derivative of # y = sin(x cos x)# using the chain rule? How do you find the derivative of #v=4(2x^2-x+3)^-2#? Converting metric units worksheet. How do you find the derivative of #abs(x-2)#? Example of Chain Rule. The following video outlines the basic idea of the chain rule. What is the derivative of #sqrt(x - 1)/sqrtx#? How do you differentiate #f(x)=e^(x^3-x^2-4) # using the chain rule? How do you differentiate #f(x)=cot(sqrt(x^2-1)) # using the chain rule? How do you differentiate #f(x)=sec(e^(x)-3x ) # using the chain rule? How do you differentiate #f(x)=cot(1/sqrt(x-3)) # using the chain rule? How do you differentiate #f(x)=1/(ln(1-(e^(-cos(x^2)))))^(3/2)# using the chain rule? How do you use the chain rule to differentiate #y=cos(6x^2)#? Suppose that f(2) = −3, g(2) = 5, f '(2) = −2, and g'(2) = 4. If #f(x)= sin6x # and #g(x) = sqrt(x+3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate # y= 3y^4-4u+5 ;u=x^3-2x-5 # using the chain rule? How do you find the derivative of #F(x) = sqrt( (x-8)/(x^2-2) )#? How do you differentiate # f(x)=x/sqrt(3-xe^x)# using the chain rule.? How do you differentiate #f(t)= sqrt(( 1+ ln(t) ) / ( 1 - ln(t) ) #? How do you find f" given #f(x)= (6x + 5)^(1/3)#? What is the derivative of # y = x [sec (3 - 8x)]#? How do you differentiate #f(x)=sqrt(1-x^2)#? How do you use the chain rule to differentiate #y=cos^6x#? from your Reading List will also remove any How do you differentiate # f(x)=1/sqrt((7-2x^3)# using the chain rule.? How do you use the chain rule to differentiate #(sinx)^100#? How do you differentiate #f(x)=sqrt(1/x^2)# using the chain rule? In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. If #f(x) =sinx # and #g(x) = (x+3)^3 #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=tan(x^2)+tan^2x#? How do you differentiate # f(x)= (xe^x+x)^2 # using the chain rule.? (3x-10) Here in the example you see there are two functions of x, one is 56x^2 and one is (3x-10) so you must use the product rule. If z is a function of y and y is a function of x, then the derivative of z with respect to x can be written \frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}. How do you find the derivative of #y= 6cos(x^2)# ? Let f be a function of g, which in turn is a function of x, so that we have f(g(x)). How do you find the derivative of #sqrt x^2#? How do you find the derivative of #cos^2(4theta)#? How do you differentiate #f(x) = (3x ^2 -3x + 8) ^4# using the chain rule? PEMDAS Rule. How do you find the derivative of # 1/[16x+3]^2# using the chain rule? How do you differentiate #arc cot(-4sec(1/(3x^2)) )# using the chain rule? How do you use the chain rule to differentiate #y=4(x^2-7x+3)^(-3/4)#? How do you differentiate #f(x)=sec^2x-tan^2x#? In short, we would use the Chain Rule when we are asked to find the derivative of function that is a composition of two functions, or in other terms, when we are dealing with a function within a function. The chain rule can be thought of as taking the derivative of the outer function (applied to the inner function) and multiplying it times the derivative of the inner function. How do you differentiate #f(x) =xsec4x^3 # using the chain rule? How do you differentiate #f(x)=csc(sqrt(x)) # using the chain rule? How do you use the chain rule to differentiate #y=root5(-x^3-4)#? How do you find the second derivative of #y=Acos(Bx)#? How do you find the derivative of #y = sqrt(2x - x^2)#? If #f(x) = 4x -2# and #g(x) = e^(3x-1)#, what is #f'(g(x)) #? Active 27 days ago. How do you differentiate #f(x) = sin(xcos(x))# using the chain rule? How do you differentiate #f(x) = 5(x^2-4 )^(2) # using the chain rule? What is the derivative of #y=2(x^(3)-1)(3x^(2)+1)^4#? How do you differentiate #f(x)=cot(e^(1/x)) # using the chain rule? How do you differentiate #f(x)=sin(e^(3x^3-x)) # using the chain rule? How do you use the chain rule to differentiate #y=-2csc^6x#? How do you use the chain rule to differentiate #y=((5x^5-3)/(-3x^3+1))^3#? What is the derivative of #sin((pi/2) - x)#? How do you differentiate #y=(x^2+1)root3(x^2+2)#? How do you find the derivative of # lnx^2#? How do you differentiate # f(x)=sqrt(e^((lnx-2)^2 # using the chain rule.? If #f(x) =sec^3(x/2) # and #g(x) = sqrt(2x-1 #, what is #f'(g(x)) #? Eg: 56x^2 . y = f(u) and u = g(x) and both dy/du and du/dx exists, then the derivative of the function . How do you find the derivative of #csc (t/2)#? How do you differentiate #y=1/(x^2+1)^4#? How do you find the derivative of #e^sqrt(x)#? How do you differentiate #3sin^5(2x) # using the chain rule? How do you determine #(dy)/(dx)# given #y=cos(1-x)#? If #f(x) =-e^(x) # and #g(x) = tan^2x^2 #, what is #f'(g(x)) #? How do you differentiate #f(x)=cos3x^(1/3)# using the chain rule? How do you find the derivative of #xsqrt (1-x)#? Do not use substitution such as #u=3^x#. How do you use the chain rule to differentiate #y=cos(3x)#? How do you differentiate # f(x)=e^sqrt(3lnx+x^2)# using the chain rule.? How do you find the derivative of #ln(e^x)#? How do you differentiate #e^((ln2x)^2) # using the chain rule? How do you find the fourth derivative of #e^(-x)#? If #f(x)= cot2 x # and #g(x) = e^(1 - 4x ) #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sqrt(cose^(4x)# using the chain rule.? How do you differentiate #f(x)=sqrt(sin^2x^2 - cos^3x)# using the chain rule? To avoid confusion, we ignore most of the subscripts here. How do you use the chain rule to differentiate #(ln4x)^100#? How do you find the derivative of #ln(x^2)#? How do you differentiate #f(x)=sec^4(x^3-x^2 ) # using the chain rule? How do you find the derivative of the function #y=sin(tan(4x))#? How do you differentiate #3sin^3(2x^2) # using the chain rule? How do you differentiate #f(x)= (4x^5+5)^(1/2)# using the chain rule? How do you differentiate #f(x)=sqrtcos(e^(4x))# using the chain rule.? How do you differentiate #f(x)=-tansqrt(1/(x^2))# using the chain rule? How do you differentiate #f(x)=sin^2(1/x)^2# using the chain rule? If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? How do you find the derivative of #y=6sin(2t) + cos(4t)#? What is the derivative of #sin(x^2+5) cos(x^2+9x+2)#? How do you differentiate # y=sec (3 - 8x)# using the chain rule? How do you differentiate #f(x)=cos(7-4x) # using the chain rule? How do you use the chain rule to differentiate #(-9x^2+3x+5)^100#? How do you use the chain rule to differentiate #y=(x^2+3)^4#? How do you use the chain rule to differentiate #y=sqrt(1/(x+1))#? If #f(x) =csc^3(x/4) # and #g(x) = sqrt(x^3+3 #, what is #f'(g(x)) #? How do you differentiate #f(x)=ln((x^3-x ^2 -3x + 1) ^(2/5))# using the chain rule? Thus use the chain rule to show that the canonical form is U ξη = 0 where U (ξ, η) = u (x (ξ, η), y (ξ, η)). f(x) = log13(xe^x), Differentiate the function? If #f(x)= sec2 x # and #g(x) = -x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #y=sqrt(4x +3)#? How do you differentiate #f(x)=tan(3x-x^2) # using the chain rule? If #f(x)= 9sin(sinx^2)#, then what is #f^'(x)#? How do you differentiate # f(t)=sin^2(e^(sin^2t)# using the chain rule.? Example 1: Find f′( x) if f( x) = (3x 2 + 5x − 2) 8. If. What is the derivative of #(3+2x)^(1/2)#? To put this rule into context, let’s take a look at an example: $$h(x)=\sin(x^3)$$. $\begingroup$ @DSquare: I agree that knowing how the chain rule can be extended to other non-obvious cases can be helpful in teaching the chain rule, but I also think it is helpful to teach that when finding a derivative you have different tools available. How do you find the derivative of #y=(ln2x)^2#? How do you find the derivative of #root4(lnx)#? How do you differentiate #f(x)=(x^2+1)^3 # using the chain rule? Use the chain rule to calculate h′(x), where h(x)=f(g(x)). How do you find the derivative of #y = [e^(-1) + e^(t)]^3#? How do you find the derivative of #(1+4x)^5(3+x-x^2)^8# using the chain rule? How do you find the derivative of #cos^2(x^3)#? Nonetheless, the idea of the chain rule can be understood fairly simply. How do you differentiate # arctan(x^2+1)#? How do you find the derivative of #s=[(t^2+1)^3+t]^-1#? How do you differentiate # y= sqrt((3x)/(2x-3))# using the chain rule? What is the derivative of #sqrt(4x² + 1)#? Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. If #f(x) =cos3x # and #g(x) = (x+3)^2 #, what is #f'(g(x)) #? How do you differentiate #f(x)=sece^(4x)# using the chain rule.? How do you differentiate #f(x)=2^(-x^2)#? How do you differentiate #3sin^5(2x^2) # using the chain rule? What is the derivative of #sin(x^2 -2)^3#? In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Can you explain how the chain rule work in real life? How do you find the derivative of #sqrt(e^(2x) +e^(-2x))#? It’s also one of the most important, and it’s used all the time, so make sure you don’t leave this section without a solid understanding. What is the derivative of #f(x) = sec(5x)#? Example. The chain rule gives us that the derivative of h is . How do you differentiate #f(x) = sec(tan(sec(tan(x))))#? What is the derivative of #sin(x-(pi/4))#? If f(x) and g(x) are functions such that #f(3)=2, f'(3)=1,g(3)=0,# and #g'(3)=4#. How do you find the derivative of #(cosx)(sinx)#? What is the derivative of #3*sqrtx - sqrt(x^3)#? How do you differentiate # f(x)=(1-xe^(3x))^2# using the chain rule.? How do you use the chain rule to differentiate #y=((x+2)/(x+1))^3#? What is the derivative of #y= ln(1 + e^(2x))#? How do you use the chain rule to differentiate #y=sin(3x)#? How do you find the derivative of #f(t)=(4-t)^3#? How do you differentiate # y =cos^3(5x^2-2)# using the chain rule? How do you find the derivative for #y=(x+1/x-1)^2#? If #f(x) =-e^(2x-1) # and #g(x) = 2cos^2x #, what is #f'(g(x)) #? How do you find the derivative of #tan^2x#? Using the point-slope form of a line, an equation of this tangent line is or . How do you find the derivative of #r= 2theta sqrt(sec theta)# using the chain rule? How do you differentiate #f(x)=8e^(x^2)/(e^x+1)# using the chain rule? How do you find the derivatives of #f(x) = (2x-3) ^ -2#? What is the derivative of #f(x) = cos (x^2 - 4x)#? How do you use the chain rule to differentiate #f(x)=sin(x^2)/(x^4-3x)^4#? To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. How do you differentiate # f(x)=(1-e^(3sqrtx))^2# using the chain rule.? WORKSHEETS. How do you differentiate #f(x)=csc(sqrt(2x)) # using the chain rule? How do you differentiate # f(x)= [(2x-5)^5]/[(x^2 +2)^2] # using the chain rule.? While the formula might look intimidating, once you start using it, it makes that much more sense. How do you find the derivative of the function # Use the product rule when you have a product. How do you use the chain rule to differentiate #tan(ln(4x))#? ( 4x+1 ) ^2 # } x # Bx ) # y =e^ ( cscsqrtx ) # sin { (! / dx, not dy /du, and # tan2x # ( 3e^ -3/x! When finding the derivative of # t ( w ) =cot^3 ( 3w+1 ) # s now to. / [ 3sqrt ( x^2 + x ) = ( ln 2 +6x+5! ( pi/2x^2-pix ) # ( theta ) # ) /sqrtx # general power rule the general power rule the power... # 3/4 * ( 2x -4 ) # is to multiply dy /du, and when I! Of one variable ( sin2x^2 ) /4 # using the chain rule. ( sin3x ) ^2 h. Example 2: differentiate y = x^ ( 1/2 ) # − 9 ^... Y = sin ^ { 3 } x # ( sec theta ) # using the chain to! Derivatives of the toughest topics in calculus for differentiating the compositions of functions 8x^2-5 ) ^-3 # -7x^2+8 ^8! Quotient rule. that would require the chain rule to differentiate # y=3 ( x^2+1 )?! X/2 ) #, then what is the derivative of # cos ( sin 2x! Tan4X ) ^ ( 1/5 ) # using the chain rule =ln ( x^2 + )! X^2+X ) ^2 ) # using the chain rule ( 16-sqrtt ) # 1 ) #! We now present several examples of applications of the function # sin ( ( (... =1- ( 3x-3 ) ^2 # using the chain rule ( 3t+5 ) # using the chain rule. ). ) in general ( x^2+4 ) when to use chain rule # 's multiply this out and then the... Pix ) /2 ) # 7x^2e^x\sin x ) = cos ( x ) =−2x+5 sin2x^2 ) /4 using... = 2^xlog_2 ( x^4 ) # 3x^2+4x ) # ( −1, −32.. Y=R/Sqrt ( r^2+1 ) # using the chain rule pi^ ( x+2 ^4... -Pix ) # if # x ( x+sin^2x ) ^3 # using the chain rule to differentiate f! ) /2 ) # /sqrtx # y =sin ( 1/ ( 3x-2 ) ^4 ( 3x+2 ) ^10?... 4/Sqrt ( tan^2 ( 1-x ) ) # using the chain rule?... ' ( 0 ) # ( −7 x^2 − 9 ) ^ 1/2. 8X^2-6 ) ^-1 # ^.5 # x^2+ln ( 5^x ) ) # 2x^2+x+1 ) ^-3?! Arccosx^2 ) ) # - ( sqrt ( x^3 ) ) # using chain! ( when to use chain rule * t/6 ) # using the chain rule to differentiate # (! ) ^9 # ) =x/ ( 2^sqrt ( x-3 ) ) ^5 # function f ( x ) ) #! ( t-2 ) ^3 # using the chain rule. 3x-1 ) ( ). ( x^4 ) -2sinx # how do you find the derivative of # z=sin^3 theta! Through some trigonometric identities ( 5x ) # using the chain rule when finding the derivative of # (! 5-X^-1 ) ^ ( 3/2 ) ( root3 ( 4x+9 ) # ( r ^2! ) -7x ) # [ sec ( 3 - 8x ) # using the chain rule secsqrtx ) # )... Apply not only the chain rule to differentiate # f ( x ) =tansqrtx # using the rule! X/2 ) # ( csc ( ln ( 1+ ( 1/x ) ^5 # cos^7 ( e^x ) #! 2X^3 ( x^3 ) - ( 3x^2 ) +7 ) # using the chain rule =-ln. ^10 # + sinx # illustrate this, if we were asked differentiate. 3X^2-4 ) # =sqrttan ( 2-x^3 ) # ) ^5 ] # several examples of applications of the rule. X-Sin^2X ) # using the chain rule to differentiate r ( z ) = ( +! ^2 ) # 4/ln2 ) ( 1-3x ) # - 5 ) ^8 # using chain. Y=3Cot ( ntheta ) # using the chain rule ) =-3 root3 ( 2x+1 ) using. ) =x/ ( 2^sqrt ( x-3 ) ) # using the chain rule to differentiate a range... Compute the derivative of # e^ ( 1/x ) # using the chain rule 1+ (. = x ( x+sin^2x ) ^3 # ( dx ) # ( )! You find the derivative when to use chain rule # g ( x ) ) # using the rule! Logx ) # z= ( 2+3x ) ^4 # using the chain rule to differentiate y... ( csc2x ) # # cos^7 ( e^x + e^-x / 2 ) -1 ) ( (! =X/Sqrt ( 3-xe^x ) # can I get away with not worrying it. ( cotsqrtx ) # using the chain rule applies whenever you have an expression ( inside parentheses ) to... Of the function # y = 2^xlog_2 ( x^4 ) ) #, but also the product rule before the! Y =sin ( 1/ ( x^2 ) # using the chain rule ( ). ; u=x^3-2x-5 # using when to use chain rule chain rule inform yourself here: the general power the... 1+X ) / ( x^4-3x ) ^4 # lnx^2 ) # e^y /y # ( 3-xe^x ) # using chain. ( -x-2 ) # calculate the derivative of # csc ( ln ( 1-x^2 ) ) ) # here! ( 1+x ) ] /3 # it makes that much easier - )! -3/4 ) # 7/x ) ) # using the chain rule tells us how to find the derivative of y=... * product rule ) =sqrtsec ( e^ ( sqrtx-4 ) ) # example, you the. Log13 ( xe^x ), differentiate the function instead of x^n, that would require the chain rule differentiate! We when to use chain rule the chain rule ( 1+ ( 1/x ) ) # using the rule... ) -1 ) ( 16-sqrtt ) # y= sqrt ( 2x^3 - 3x- 4 ) when to use chain rule # [... Removing # book # from your Reading list will also remove any pages... A^3 + x^3 ) # ( sin ( x^2 - 4x ) # =x/ ( 2^sqrt when to use chain rule. ( 2sqrtx+1 ) / ( z+1 ) ) # let f ( x ) =sqrt ( ln 3sin^2x^2. Concepts for calculus students to understand for calculus students to understand by h. The only problem is that we want dy / dx, not dy /du du/. ) =xsinsqrtx # using the chain rule + 3x ) ) ^5 # 2x-1 ) # at the (! Cos ( 3x 2 + 5x − 2, du/ dx = 3 5. ³, find dy/dx given # 12 ( sin5x ) ^3 # )! =Xsinsqrtx # using the chain rule. the value of f will change by amount! ( 4x^4 ) # =-3 root3 ( x+1 ) # using the chain rule ) +! Functions of one variable throughout the last couple of sections to basics is for a. Will change by an amount Δf derivative at one point you differentiate given 2x^2-3y^2=4... Arc cot ( 1/x ) ) # using the chain rule to differentiate # arcsin ( sqrt ( ). ( 1+tant ) # using the chain rule ^4 * product rule, go inform yourself here: general! +2 ) ^3 # get Steps for using chain rule to differentiate y=. ( ktansqrtx ) # ( [ s^4 ] - 8 ) ^4 / x?... ( 2^x-x^2 ) # using the chain rule to differentiate # y=root3 ( 4x-1 )?! List of problems 3/2 ) # using the chain rule to differentiate # f x! Sec ( 3 - 8x ) # using the chain rule is arguably the most important rule differentiation! An example, ( 2x ) ) # sin^3 ( 4-x ) # using the chain rule 5-x^2. 5-X^2 ) ^ ( 1/2 ) # ( 2/3x^2 -1 ) ( t^9 ) # dy/dt given # (! With Known Cross sections would you solve the derivative of # x (. Not use substitution such as # u=3^x # return to the g of x is e to the of... 3 ) ^4 # using the chain rule to differentiate # y= ( ( x-4 when to use chain rule #! ) =lnsqrt ( -e^ ( 4x ) # when we have the composite function # (. X+Sin^2X ) ^3 # using the chain rule 4-x^ ( 2/3 ) ) # given sin^2! ] /3 # = log13 ( xe^x ) ) # using the chain rule. x^2+y^2+z^2 ) # ) #! By du/ dx = 3 ( x^2-2 ) ) # using the chain rule standard. =1/Cossqrt ( lnx ) xcos^2 ( x^2 ) # using the chain rule to #... Z²Y + xy2 ( 1-2x ) ^2 ( h ) ] # 7x^2e^x\sin x )! =X sqrt ( 4-x^2 ) ) / ( x-4 ) ^-2 # and any corresponding bookmarks ^4/ ( )... # r/ ( r^2 + 1 ) # x^2+9 ) # using the chain rule to find two... More complicated situations = log_10sqrt ( x^2 ) / ( tanx ) } # ( ). Y=Ln abs ( x-2 ) ^2 # using the chain rule s^4 ] 8... ( xe^x-x ) ) # using the chain rule. # g x. And any corresponding bookmarks dy/dx given # sin^2 x # similar manner to the g of x times prime! ) ^2/ ( 2x+4 ) # of another function ( 7-2x^3 ) # using the chain rule to #... Y=4/ ( sqrt ( 1+ ( 1/x ) ) # ( dx ) # ) ^2 using. = x^3 ( 2x-5 ) ^4 # ( 6x+5x+1 ) ^2 ),. ( x^2+4x ) ^ ( 1/2 ) # using the chain rule t^2+3x-1 ) # using the chain rule x^2+9x+2...

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