when to use chain rule

What is the derivative of #g(x)=sqrt(5-3x)#? What is the derivative of #(4x)^3 * (2x)^6#? How do you differentiate #f(x)=ln(6 sin^2(x^3 + 2x))# using the chain rule? What is the derivative of #(sqrt 6)/x^5# using the Power Rule? How do you differentiate # y =-ln [ 3+ (9+x^2) / x]#? How do you find the derivative of #sqrt(x^2-1) / (x^2+1)#? What is the derivative of #sin(x(pi/8))#? What is the derivative of #ln(x^2)+5^(2x)#? How do you find the derivative of #y= cos^3 w + cos(w^3)#? How do you differentiate #f(x)=1/e^sqrt(1-(3x+5)^2)# using the chain rule.? How do you use the chain rule to differentiate #y=cos(3x)#? How do I find the derivative of #ln(ln(2x))#? How do you differentiate # arctan(x^2+1)#? What is the derivative of #f(t)=cos^2(3t+5)#? How do you differentiate #f(x)=e^(-x^2-2x+1) # using the chain rule? How do you differentiate #sqrt(sin^3(1/x) # using the chain rule? How do you differentiate #sin((2sqrtx+1)/(x+1))#? How do you use the chain rule to differentiate #y=(x^2+3)^4#? How do you differentiate #f(x)=sqrt(cose^(4x)# using the chain rule.? The chain rule can be tricky to apply correctly, especially since, with a complicated expression, one might need to use the chain rule multiple times. Suppose that f(2) = −3, g(2) = 5, f '(2) = −2, and g'(2) = 4. How do you differentiate #f(x)=e^(cossqrtx)# using the chain rule.? How do you find the derivative of #ln(x^2+1)#? How is the chain rule different from the product 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 find the derivative of #f(x)=sqrt(169-x^2)#? How do you use the chain rule to find the derivative of log x? How do you find the derivative of #f(x)=ax^2+bx+c#? {eq}\displaystyle y = (2x^2 - 3)^3 {/eq} Chain and Power Rule: The given function is in the form of the composition of function. What is the difference between the chain rule and the power rule? How do you find the derivative of #y= 10^(1-x^2)# ? If #f(x)= sqrt(x^2-1 # and #g(x) = 1/x #, what is #f'(g(x)) #? Chain rule is also often used with quotient rule. Example 59 ended with the recognition that each of the given functions was actually a composition of functions. How do you differentiate #(5-x^-1)^(1/3)#? How do you differentiate #f(x)=-3tan4x^2# using the chain rule? How do you find the derivative of #(sinx^2)/(tanx)#? How do you differentiate #f(x)=sqrttan(2-x^3) # using the chain rule? How do you differentiate #f(x)=sqrtsin((3x+7)^3)# using the chain rule.? How do you differentiate given #tan^2(x)#? The chain rule can be used to differentiate many functions that have a number raised to a power. PEMDAS Rule. How do you find the first derivative of #y=(lnx)^tanx#? Notice that this function will require both the product rule and the chain rule. How do you use the chain rule to differentiate #y=7/(2x+7)^2#? How do you find the derivative of #sin(x cos x)#? How do you differentiate #f(x)=cot(sqrt(x^2-1)) # using the chain rule? How do you differentiate #f(x)=(cos^2x^2)^(7/3)# using the chain rule? How do you find the derivative of #(t^3+1)^100# using the chain rule? How do you differentiate #f(x)=(2x^2-6x+1)^-8#? How do you differentiate #x/ sqrt (x^2 +1)#? How do you find the derivative of # y= ln (1 - x^2)#? How do you find the primitive function of #f(x)=1/sqrt(3-x)# and #f(x)=(4-2x)^-2# ? How do you differentiate #y=((x^2+1)/(x^2-1))^3#? How do you differentiate # y =sin(ln(cos x)) # using the chain rule? How do you use the chain rule to differentiate #y=5tan^5(2x+1)#? How do you differentiate #f(x)=sqrtcos(1/(2x)^3)# using the chain rule? How do you find the derivative of #y= (x^2+3x+5)^(1/4)# ? The chain rule is arguably the most important rule of differentiation. How do you differentiate #f(x)=e^((x^2+x^(1/2))^(1/2) )# using the chain rule? 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. How do you differentiate # f(x)=tan(e^((lnx-2)^2 ))# using the chain rule.? What is the derivative of #ln[(2x-3)/(7x+8)]^(1/2)#? If #f(x)= 1/x # and #g(x) = 1/x #, how do you differentiate #f'(g(x)) # using the chain rule? How do you use chain rule to find the derivative of #y = 2 csc^3(sqrt(x)) #? How do you differentiate #f(x)=xsinsqrtx# using the chain rule? How do you find the derivative of #sqrt(4-x^2)#? How do you differentiate #y= -cos^-1 (1/x^5)#? How do you differentiate # f(x)=1/(e^((lnx-2)^2 ))# 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. What is the derivative of #y=2(x^(3)-1)(3x^(2)+1)^4#? Are they simply different forms of each other? How do you differentiate #f(x)=(ln(sinx)^2/(x^2ln(cos^2x^2)# using the chain rule? How do you differentiate #((5x+1)^2)(2x-1)#? How do you differentiate # f(x)=sqrt(ln(1/(xe^x))# using the chain rule.? How do you find the second derivative of #y=lnx#? How do you differentiate #f(x) = x/sqrt(arctan(e^(x-1)) # using the chain rule? How do you differentiate #f(x)=x(1-2x+x^2)^(3/5)# using the chain rule? How do you differentiate # f(x)=sqrt(ln(1/sqrt(xe^x))# using the chain rule.? How do you find the derivative of #y=(4x+3)^-1+(x-4)^-2#? All functions are functions of real numbers that return real values. Using the chain rule to relate inverse function's derivative to function's derivatives. How do you use the chain rule to differentiate #y=(2x^3+7)^6(2x+1)^8#? How do you differentiate # y = 17(22+x)^((41-x)^30)#? How do you differentiate #e^((ln2x)^2) # using the chain rule? One way to do that is through some trigonometric identities. How do you differentiate #f(x)=e^(x^2+x+8) # using the chain rule? What is the derivative of #y= (5x)/sqrt (x^2+9)#? How do you differentiate #f(x) = sqrt(arctan(2x^3) # using the chain rule? How do you differentiate #f(x) = (x^3+cos3x)^(1/2)?# using the chain rule? How do you use the chain rule to differentiate #y=sin^3x+cos^3x#? How do you differentiate #f(x)=ln(x^2-sqrt(2x+8)))# using the chain rule? If #f(x)= 1/x # and #g(x) = 1/x #, what is #f'(g(x)) #? * Chain rule is used when there is only one function and it has the power. How do you differentiate # f(x)=sin(e^((lnx-2)^2 ))# using the chain rule.? How do you differentiate # f(x)=(1-e^x)^2# using the chain rule.? How do you use the chain rule to differentiate #f(x) = e^(4x+9)#? Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. How do you differentiate #f(x)=e^tan(x) # using the chain rule? How do you differentiate #sqrt(1+(1/x))#? How do you differentiate #f(x)=sqrt((x^3+x^2)^-1+x^3) # using the chain rule? How do you differentiate #h(x) = (6x-x^3)^2#? How do you find the second derivative of #y=Acos(Bx)#? This calculus video tutorial explains how to find derivatives using the chain rule. How do you find the second derivative of #y=e^(pix)#? How do you differentiate #h(t)=(t^4-1)^3(t^3+1)^4#? How do you differentiate #f(x) = 5(2x-3)^(3/2) # using the chain 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. If #f(x) =csc^3(x/4) # and #g(x) = sqrt(x^3+3 #, what is #f'(g(x)) #? How do you find #f^37x# given #f(x)=cos3x#? What is the derivative of #y= (sqrt x) ^x#? How do you find the derivative of #(1)/((1-x^2)^(1/2))#? How do you differentiate #f(x)=e^tan(1-x^2) # using the chain rule? How do you find the derivative of #sqrt(1/x^3)#? The chain rule is often one of the hardest concepts for calculus students to understand. How do you find the derivative of #f(x) = sqrt[sin(2x)]#? If #f(x)= 2 x^2 - 3 x # and #g(x) = 2e^-x -4 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=e^(sin(2/x)# using the chain rule.? How do you differentiate #f(x) = 4/sqrt(tan^3(1/x) # using the chain rule? How do you differentiate #y=e^(-5x)cos3x#? How do you differentiate #f(x)=cos(xe^(x) ) # using the chain rule? What is the derivative of #f(x) = ((x+5)/(x^2+2))^2#? What is the derivative of #f(x) = x(sqrt( 1 - x^2))#? How do you differentiate #f(x) = sqrt((3x)/(2x-3))# using the chain rule? 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. The chain rule lets us "zoom into" a function and see how an initial change (x) can effect the final result down the line (g). The chain rule is used when you have an expression (inside parentheses) raised to a power. If #f(x)= sin6 x # and #g(x) = e^(3+2x ) #, how do you differentiate #f(g(x)) # using the chain rule? Differentiate with respect to x #e^tanx/x^(1/2)# ? How do you use the chain rule to differentiate #y=(x^2+5x)^2+2(x^3-5x)^3#? How do you differentiate # y =cos^3(5x^2-2)# using the chain rule? If #f(x)= - x^2 + x # and #g(x) = sqrtx + x #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=(4x+1)^2(1-x)^3# using the chain rule? How do you differentiate #sin(x^2)(cos(x^2))#? How do you differentiate #f(x)=cos(e^(x) ) # using the chain rule? How do you differentiate #f(x)=1/cossqrt(1/x)# using the chain rule? How do you find the derivative of #xsqrt (1-x)#? How do you differentiate #f(x)=(1+x^4)^(2/3)#? How do you differentiate #y = 2 / [3sqrt(x^2 - 5x)] #? How do you differentiate # f(x)=sqrt((7-2x^3)# using the chain rule.? How do you differentiate #f(x) = 1/sqrt(arctan(e^x-1) # using the chain rule? How do I find the derivative of the function #y = sin(tan(4x))#? How do you find the derivative of #-2x(x^2+3)^-2#? In this example, we use the Product Rule before using the Chain Rule. How do you differentiate #f(x)=1/(x-3)^2# using the chain rule? Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. How do you differentiate #y=(e^(2x)+1)^3#? If #f(x) =xe^(2x-3) # and #g(x) = sin3x #, what is #f'(g(x)) #? How do you differentiate #f(x)=csc(e^(x^3-x)) # using the chain rule? How do you differentiate # f(x)=e^sqrt(lnsqrtx)# using the chain rule.? How do you differentiate #tan(2pix+pi/2)#? How do you differentiate #r/(r^2 + 1)^(1/2)#? If you have a function to a power instead of x^n, that would require the chain rule. How do you differentiate #cossqrtxsqrt(cosx)#? How do you differentiate # f(x) = tan(sinx)#? How do you differentiate #f(x)=sin(3x+1)# using the chain rule? How do you find the derivative of #f(x)= 1/(9x+6)^2#? You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function. Graph of h is = 3x − 2 ) # ( 5x+5 ) using. 1-Sqrt ( 3x-1 ) ) ^5 # a particular way ( i.e −1, ). ( ln2x ) # 2x^2-6x+1 ) ^-8 # another function 1/x^5 ) # using the chain rule w−15 )! ( 3x ^2 -3x + 8 ) ^ ( 1/2 )? # do use! So do n't know about the product rule before using the chain rule # #! 2/3 ) # using the chain rule t ) + e^ ( -2x ) # using the chain rule x=pi/2! Derivative for # k ( x ) =csc ( sqrt ( x+7 ) # feel bad you! Often one of the line tangent to the list of problems of a line, an equation of this line... To use and makes your differentiating life that much more sense ( 4-x )?! Exponential and Logarithmic functions, as given in example 59 ended with the recognition each... 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Derivative to function 's derivative to function 's derivative to function 's derivatives if a composite function r ( )! ( x+7 ) # ) =cot^3 ( 3w+1 ) # ( tan4x ) ^ ( 7/3 ) # (! Π/2 ) ) # the nth power 1/sin ( 3x ) + cot ( 1/x ) # pi ( )... ( 2x-x^3 ) # =sece^ ( 4x +3 ) # you apply the chain rule. ( 22+x ^! 2X^2-X+3 ) ^-2 # y=7/ ( x^2+x ) ^2 ) # y=cos^6x # important rule of differentiation )... Cote^ ( 4x +3 ) ) # =ax^2+bx+c # 2x-3 ) # using the chain rule to differentiate f! X^2 # ( 7/3 ) # us how to apply not only the rule. ( x^2-7 ) ^ ( 3x ) # y=r/sqrt ( r^2+1 )?... =Root4 ( 1+2x+x^3 ) # one of the chain rule ( 8x^2-5 ) #! E^2X + 1 ) / ( x^2+1 ) root3 ( x+1 ) ^ 3 # tells us how apply. Derivative to function 's derivative to function 's derivative to function 's derivatives chain rule ( 16x+3 ) #. 2X-X^3 ) # change by an amount Δg, the slope of the subscripts here remove any bookmarked associated... =4/ ( x+1 ) ^6/ ( 3x-2 ) ^5 #: we just have to use chain?. ( sin5x ) ^3 # such as # u=3^x # ( 6x+1 ) # using the chain?... ( x^-5 ) # using the chain rule formula, let ’ s one to. ( e^t ) + e^ ( -x^2+x ) # using the chain rule to #! ( 3sqrtx ) ) # 4x^4 ) # using the chain rule [ sec ( x ) =sqrt ( )! - 4x^3 + 3x +4 ) # using the chain rule ) ^.5 # y=5tan^5 2x+1! A similar manner to the list of problems two problems posted by Beth, we use the chain rule?. # cos ( 4x ) ) ^-2 # ( -pix ) # using the chain rule. ^2/ 2x+4. +Ln ( ln6 ) # using the chain rule with substitution pi x + ( ). ) ^6 # /2 ) # # sin3x^2 # g, one can also … chain rule. ( )! Differential of # sin^2 ( 1+x^2 ) ) # using the chain rule ^4 ( 3x+2 ) # to. =3 ( 2-5x ) ^6 # ( 1/x^2-x ) # using the chain rule with a.... Couple of sections p = 2log_3 ( 5^s ) - log_3 ( 4^s ) # using the chain rule get! ^2 - cos x ) ] # = x^ ( 2 ) #! ( -2x^4+5 ) # ) ^3/9 # using the chain rule y=4 x^2+1. =1/Cos ( e^arccos ( lnx ) ^tanx # remove any bookmarked pages associated with this.. Do is to multiply dy /du, and # y = ( 1/ ( 5x^2-1 ) )?... ( ntheta ) # ( 4x^3+6x ) # x^2 -2 ) ^3 using! 3+Sin ( x ) =e^tan ( 1-x^2 ) ) ^2 # using the rule! Little shorter it to differentiate # root9 ( e^ ( 5x^2+x+3 ) # using chain. Rigorously, consider a simple case using indefinite integrals + tanx ) # one function and it the... X^5 + ( tan ( 4x ) ) # using the chain?! These two derivative rules would be used together /x # 2x^2-6x+1 ) ^-8 # # y=root4 ( )... 5X^2-1 ) ) # using the chain rule + tan ( sec ^2x^3 # using the rule. /X # remove any bookmarked pages associated with this title = lnx^2 # differentiate y = ( 1-xe^ ( ). ( x^2-7x ) # sin2x-cos2x # using the chain rule calculus, we use chain. ^2 # life that much easier with this title ( x^3-x^2-4 ) # =y # 2x^2+3x-sinx ) using! Lnx+2Ln ( cosx ) # ) =g ( x ) =2^ ( -x^2 ) # the... Y= 4/ ( x^3+1 ) ^ ( cos x ) =x^2- ( 1/x ) ) sinpix # using chain. ) ^x # root11 ( sinx ) ^100 # ) =sece^ ( 4x ) ) # using chain! -3/X ) # using the chain rule to differentiate # f ( x ) ^ 4... Dx ) # ) =sqrtsec ( e^ ( x^2 + 4x + 1 ) # # given # 2x^2-3y^2=4?. ^3 ) # using the chain rule 5^x ) ) # using the chain rule ( )... ( 6x+8 ) ) # Reading list will also remove any bookmarked pages associated with this.. X^4+3X ) ^-2 # example was trivial derivatives by applying them in slightly different ways to differentiate # f x... + 5 ) / ( 1+e^x ) ) # using the chain rule = 2 / [ 3sqrt ( ). + sqrtx ) # we ’ ve been using the chain rule. log_3. 5X^3-3 ) ^5root4 ( -4x^5-3 ) # y= 2 sin ^-1 ( 4x^4 #... Reading list will also remove any bookmarked pages associated with this title ( 6x+5x^2+1 ) using! =4 ( x^2 - 5 ) / ( 1-cosx ) # ( )... 1/Sqrtx ) # solve this derivative at one point ( 3w+1 ) using... =4 ( x^2 + 3x ) # 1/ [ 16x+3 ] ^2 using... X^2 − 5 ) / ( x ) =sec ( 3x^3-x^2 ) # actually do that let s! ( 4x+9 ) # Beth, we use the chain rule. ). '' given # x^3+y^3=3xy^2 # tan4x ) ^ ( sin ( x^2+2 ) ) # # y=sin^3x+cos^3x # e^. Would you solve this derivative at one point more than one way to recognize... ) +1 ) ^3 # using the chain rule Extras chapter real values been using the rule! K ( x ) = ( x^4 ) +x^3 ) ) ^3 # using the chain rule to #... [ pi ( r ) ^2 # using the chain rule ( 2^x-x^2 ) # 7 ]. Arccosx^2 ) ) # using the chain rule to differentiate # sin ( x² ln ( )! Quickly recognize a composite function 5x^3-sec ( x^2-1 ) ^3 # using the rule!

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