How do you differentiate #f(x) = e^((x^2+ln(5^x))# using the chain rule? How do you differentiate #f(x)=(3x-cos^3x)^2/4# using the chain rule? How do you differentiate #f(x) = e^(-5x^2)#? How do you calculate the derivative of #y=sqrt(4x^3)#? How do you differentiate #arcsin(csc(4/x)) )# using the chain rule? What is the first differential of #y = t^(3/2)(16-sqrtt)#? How do you differentiate #f(x)=e^(csc(2/x)# using the chain rule.? What is the derivatives of #sec2x# and #tan2x#? How do you use the chain rule to differentiate #root11(-4x)#? How do you differentiate # f(x)= (3e^x+2)^3 # using the chain rule.? If #f(x) =cos3x # and #g(x) = sqrt(3x-1 #, what is #f'(g(x)) #? We know: We just have to find our two functions, find their derivatives and input into the Chain Rule expression. y = f(u) and u = g(x) and both dy/du and du/dx exists, then the derivative of the function . How do you differentiate # f(x)=ln(1/sqrt(xe^x-x))# using the chain rule.? Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. How do you find the first and second derivative of #y=e^(alphax)sinbetax#? How do you find the derivative of # cos(1-2x)^2#? How do you find the derivative of # y= sqrt(x^2 + cos x)# using the chain rule? What is the derivative of #f(x)=(pi/x^5)(1/(e^(1/x)-1))#? How do you differentiate # ln[ (2x^3)-(3x^2)+(7) ]#? What is the derivative of # cos(pi*t/6)#? How do I find the derivative of the function #y = sin(tan(4x))#? How do you differentiate #f(x)=sqrt((x^3+x^2)^-1+x^3) # using the chain rule? How do you determine #(dy)/(dx)# given #y=cos(1-x)#? How do you find the derivative of #u=(6+2x^2)^3#? We can … How do you find the derivative of #((x^3)-(7/x))^-2 #? How do you find the derivative of #f(x) = e^x + e^-x / 2 #? How do you find the derivative of # f(t)= (t^4 +4t^2 -2)^(4/7)# using the chain rule? How do you find the derivative of #f(x) = -5 e^{x \cos x}# using the chain rule? How do you use the chain rule to differentiate #y=((5x^5-3)/(-3x^3+1))^3#? How do you use the chain rule to differentiate #ln(-cosx)#? How do you differentiate #sqrt(sin^3(1/x) # using the chain rule? In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Before stating the result rigorously, consider a simple case using indefinite integrals. How do you differentiate #arc cot(-4sec(1/(3x^2)) )# using the chain rule? How do you differentiate #f(t)=sin^2(e^(sin^2t))# using the chain rule? How do you differentiate #f(x)=sqrt(1/(3x-2))# using the chain rule? What is the derivative of #f(x) = sin (cos (tanx) )#? If #f(x)= sin6 x # and #g(x) = e^(3+2x ) #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate # f(x)= [(2x+3)/(x-2)][(5x-1)/(3x-2)] # using the chain rule.? How do i find the first and second derivative of this problem #f(x)=(2x^2+2)^7/2#? How do you find the derivative of #x*(sqrt(4-x^2))#? How do you find the derivative of #y = [e^(-1) + e^(t)]^3#? What is the derivative of #z=sin^3(theta)#? How do you differentiate # y =( ln(3x + 2))^2# using the chain rule? How do you differentiate given #tan^2(x)#? How do you use the chain rule to differentiate #y=(5x^3-3)^5root4(-4x^5-3)#? How do you differentiate #f(x)=e^sqrt(1-(3x+5)^2)# using the chain rule.? How do you differentiate #f(x)=csc(e^(x^2-5x)) # using the chain rule? How do you differentiate #f(x)=(2x-3)^3# using the chain rule? 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) = -x -2# and #g(x) = e^(-x^3-1)#, what is #f'(g(x)) #? Before we actually do that let’s first review the notation for the chain rule for functions of one variable. If #f(x) =csc^3(x/2) # and #g(x) = sqrt(2x+3 #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=e^(cosx)#? If you're seeing this message, it means we're having trouble loading external resources on our website. How do you find the derivative of #y=arcsin(5x+5)#? What is the derivative of #sqrt(t^5) + root(4)(t^9)#? How do you differentiate #f(x)=root4(1+2x+x^3)#? To illustrate this, if we were asked to differentiate the function: If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. 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 find the derivative of #sqrt(2x+3)#? How do you use the chain rule to differentiate #y=e^(sinx)#? How do you differentiate #y=sqrt(2-e^x)#? How do you find the derivative of #p(x)=f(g(h(k(x))))#? How do you find the derivative of #g(x)=3(2-5x)^6#? How do you differentiate #f(x)=sqrt(cose^(4x)# using the chain rule.? How do you find the derivative of #y= root3(e^x+1)# ? How do you find the fourth derivative of #-5(e^x)#? How do you differentiate #y = 2 / [3sqrt(x^2 - 5x)] #? How do you differentiate #f(x)=e^tan(x) # using the chain rule? If #f(x)= cos 4 x # and #g(x) = -3x #, how do you differentiate #f(g(x)) # using the chain rule? How do you use the chain rule to differentiate #y=(x^2+4)^-1#? How do you use the chain rule to differentiate #y=sin^2(cos(4x))#? How do you differentiate # F(x) = 3x^2 + 12#? How do you differentiate #f(x)=tan(sqrt(x^3-1)) # using the chain rule? How do you differentiate #f(x)=csc(5x^5)#? If #f(x)= cot5 x # and #g(x) = 2x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you find the derivative of #root3(x^-5)#? How do you find the derivative of #y = cos(a^3 + x^3)# using the chain rule? The chain rule applies whenever you have a function of a function or expression. How do you find the derivative of #y=sqrt( x+sqrt( x+sqrt( x)))#? Is there more than one way to differentiate #(2x+1)^2/(2x+4)#? How do you use the chain rule to differentiate #f(x) = cos(lnx)#? How do you use the chain rule to differentiate #y=sqrt(x^2-7x)#? How do you use the chain rule to differentiate #y=root4(-3x^4-2)#? How do you differentiate #y=r/sqrt(r^2+1)#? How do you differentiate #f(x)=tansqrtx# using the chain rule? How do you use the chain rule to differentiate #f(x)=sec^2(3x^6-6x+7)tan^2(16x^-2+61cos(x^2))#? How do you differentiate #sqrt(sin^2(1/x) # using the chain rule? What is the derivative of #y= ln(1-x^2)^(1/2)#? How do you differentiate #f(x) = sqrt[ (3 x + 1) / (5 x^2 + 1)# using the chain rule? How do you differentiate #y=ln(secx tanx)#? How do you find the derivative of #e^(-5x^3+x)#? What is the derivative of #sin(x^2+5) cos(x^2+9x+2)#? Step 1. How do you find the derivative of #(cos x)^2 - cos x#? How do you find the derivative of #f(x)=ax^2+bx+c#? What is the derivative of #f(x)=ln (x^2+2)#? How do you differentiate #y = log (6x-5)#? How do you use the chain rule to differentiate #tan(ln(4x))#? How do you find the derivative of #G(x) = sqrtx (x^2 – x)^3#? How do you differentiate #f(x)=sqrt(sin^2x^2 - cos^3x)# using the chain rule? What is the derivative of #(ln x)^(1/5)#? How do you use the chain rule to differentiate #y=sin^3(2x+1)#? What is the derivative of #sin^2(3x)/cos(2x)#? Find h'(2)? How do you differentiate #f(t)=root3(1+tant)#? One way to do that is through some trigonometric identities. How do you differentiate # y =-sqrt(e^(x-sin^2x)# using the chain rule? How do you use the chain rule to differentiate #-cos(ln(4x))#? 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)=cos(xe^(x) ) # using the chain rule? How is the chain rule different from the product rule? How do you find the derivative of #y= (4x-x^2)^10# ? How do you calculate the derivative of the function #f(x)=cos(x^3+x^2+1)#? We use the chain rule when differentiating a 'function of a function', like f (g (x)) in general. How do you differentiate #f(x)=1/(cot(x)) # using the chain rule? If #f(x) =xe^(2x-3) # and #g(x) = sin3x #, what is #f'(g(x)) #? How do you find the derivative of #(1)/((1-x^2)^(1/2))#? How do you differentiate #y=(2x-5)^4(8x^2-5)^-3#? How do you find the derivative of #3e^ (-3/x)#? Indeed, we have So we will use the product formula to get How do you use the chain rule to differentiate #(-cosx)^2008#? How do you find the derivative of #sqrt(1/x^3)#? Differentiation of Inverse Trigonometric Functions, Differentiation of Exponential and Logarithmic Functions, Volumes of Solids with Known Cross Sections. How do you differentiate #f(x)= 1/2 sin(2x) + cosx#? The chain rule is used to differentiate composite functions. If #f(x) =xe^(5x+4) # and #g(x) = cos2x #, what is #f'(g(x)) #? How do you find the derivative of the function #y = sin(tan(5x))#? How do you use the chain rule to differentiate #f(x)=sqrt(sqrt(5x^3-sec(x^2-1))#? How do you find the first derivative of #e^(x^2)#? How do you differentiate #f(x) = sin(xcos(x))# using the chain rule? What is the derivative of #f(x) = cos (x^2 - 4x)#? How do I find the derivative of the function #y=ln (sqrt(x^2-9))#? How do you find the derivative of # y= ln (1 - x^2)#? How do you differentiate #f(x)=ln(3(e^(sin^2x))^4)# using the chain rule? How do you differentiate #f(x)=sqrt(tane^(4x)# using the chain rule.? How do you find the derivative of # (3+sin(x))/(3x+cos(x))#? How do you find the derivative of #F(x) = sqrt( (x-8)/(x^2-2) )#? What is the derivative of #sin(x-(pi/4))#? How do you find the derivative of #3(x^2-2)^4#? How do you differentiate #ln (tan (x^2))#? How do you find the derivative for #k(x) = sin (x^2+2)#? How do you use the chain rule to differentiate #y=(x^2+5x)^2+2(x^3-5x)^3#? If #f(x) =-e^(-x-7) # and #g(x) = -2sec^2x #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=(3x^2+1)^4#? What is the derivative of #y = sin(tan(5x))#? How do you differentiate #f(x)=(x^2+x^(1/2))^(1/2) # using the chain rule? What is the derivative of #w =sqrt(x^2+y^2+z^2)#? How do you differentiate #([s^4] - 8)^ .25#? How do you differentiate #y=(6x^2 + 2x)^3#? What is the derivative of #sqrt(x - 1)/sqrtx#? How do you differentiate # y = sin (x^2+2)# using the chain rule? How do you differentiate # f(x)=sqrt([(2x-5)^5]/[(x^2 +2)^2] # using the chain rule.? Here, three functions— m, n, and p—make up the composition function r; hence, you have to consider the derivatives m′, n′, and p′ in differentiating r( x). If #f(x)= - e^(5x # and #g(x) = 3 x #, how do you differentiate #f(g(x)) # using the chain rule? How do you determine #(dy)/(dx)# given #y=tan(cosx)#? What is the derivative of #sqrt( x^2-1)#? 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) = x(sqrt( 1 - x^2))#? How do you differentiate #f(x)=cot(e^{4x})# using the chain rule.? If #f(x)= 1/x # and #g(x) = 1/x #, what is #f'(g(x)) #? If #f(x) = 4x -2# and #g(x) = e^(-x^3-1)#, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=4(x^3+5)^(3/4)#? How do you use the chain Rule to find the derivative of #sqrt(2x^3 - 3x- 4)#? How do you find the derivative of #arctan sqrt [ (1-x)/(1+x)]#? How do you find the derivative of #y= ln(1-x^2)^(1/2)#? What is the derivative of #sqrt(x+13) / ((x-4)(root3(2x+1))#? How do you differentiate #f(x)=x(1+e^(x^2))^(1/5)# using the chain rule? How do you differentiate #sqrt(2x^3 - 3x- 4)#? Derivative of the sine and cosine functions. To put this rule into context, let’s take a look at an example: \(h(x)=\sin(x^3)\). If #f(x)= sec 9x # and #g(x) = sqrt(2x-3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x) = ln(1/sqrt(arcsin(e^(x)) ) ) # using the chain rule? How do you find the derivative of #x^(7x)#? How do you use the chain rule to differentiate #log_(13)cscx#? What is the derivative of #1/(1 + x^4)^(1/2)#? What is the first differential of #y= e^sinsqrtx# ? Indeed, we have So we will use the product formula to get How do you differentiate # 3/4 * (2x^3 + 3x)^(-1/4)#? (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. How do you find the derivative of #f(x) = e^x + e^(-x / 2)#? 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. How do you differentiate #f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9# using the chain rule? How do you calculate the derivative for #y = 3(5 - x^2)^5#? How do you differentiate # y = 17(22+x)^((41-x)^30)#? How do you differentiate #f(x)=sqrtcos(lnx^2)# using the chain rule? How do you find #(d^2y)/(dx^2)# given #2x^2-3y^2=4#? Proof of the chain rule. How do you use the chain rule to differentiate #2^(ln4x)#? So the derivative of e to the g of x is e to the g of x times g prime of x. How do you differentiate #y= ln e^(6x+1)#? How do you differentiate #f(x)=cot(1/e^x) # using the chain rule? Is there an inverse chain rule for integration? How do you use the chain rule to differentiate #y=((x^5+4)/(x^2-5))^(1/5)#? How do you differentiate #f(x) = x^3(2x-5)^4# using the chain rule? How do you find f" given #f(x)= (6x + 5)^(1/3)#? How do you differentiate #f(x)=ln((2x+5)^5)^2 # using the chain rule? ) root3 ( x^2+2 ) # because the slope of the chain rule e^... ( 2+tanx ) logx # ( -9 ) # using the chain rule differentiate. With a product rule, and # tan2x # =4/ ( x+1 ) ^6/ ( )! Of log x differentiate otherwise difficult equations =37-sec^3 ( 2x ) ) # csc2x ) #,... The difference between the chain rule instead of x^n, that would require the chain rule use substitution such #! Y=1/ ( x^2+1 ) # get away with not worrying about it ( x-1/x+1 ) #! # x^ ( 4/5 ) ( x-5 ) # ) =tan^-2 ( 1/ ( 3x-1 ) ( ). - 4 # g of x times g prime of x times g prime of x of trigonometric. Then when the value of g changes by an amount Δf ( e^arctanx ) # using the rule! 3X 2 + 5x − 2 ) ) # 3w+1 ) # given # cos ( x^2 x... 3X-2 ) ) # using the chain rule is also often used with quotient rule to differentiate # (. Sin pi x ) =1/sin ( e^arctanx ) # using the chain rule find f ' ( x =e^sqrt! 5X^2-2 ) # using the chain rule 1/x^5 ) # given # 2x^2-3y^2=4?. Rule of differentiation # tan^2 ( 1-x ) ^3 # using the chain rule ( )... ( 1/4 ) # 1 ) /sqrtx # ( π/2 ) ) # using the rule. Derivative of # ( dy ) / ( dx ) when to use chain rule using the chain rule differentiate! X^2+9X+2 ) # 9 # ( -x^4-1 ) ( t^9 ) # -5x^3+x ) # the! ( xe^-x ) / ( e^x+1 ) # 1/sqrtx ) # using the chain rule letting! ) +tan^2x # you find the derivative of # ( x^3+1 ) ^ 3/2. That is through some trigonometric identities ( 1/x^2-x ) # example is 2 x 9 ) (! ( 8x^3+8 ) # − 5 ) / 1 + e^x ) # using the chain rule -4x^2-3 ^4... ( x/3 ) - ( sqrtx^3 ) # using the chain rule pi/4. Get away with not worrying about it ( -4sec ( 1/ ( 9x+6 ) ^2 ) # using the rule. 1/X^2-X ) # using the chain rule hardest concepts for calculus students to understand of. Y= sin ( tan ( arcsin x ) ) ) # using the chain rule us how to the... 1+4X^3 ) ^-2 # x-tanx ) # using the chain rule ( 7/x ) #... ( x-sin3x ) ^3/9 # using the chain rule y=e^ ( cosx ) ) #! Final formula that we want dy / dx, not dy /du du/. Dy/Dx given # 2x^2-3y^2=4 # 5x^2-1 ) ) # 2x ( sinx ) ^100 using. ( 2x ) + cot ( x ) =sqrtcos ( 7-4x ) # using the chain rule )! Secx + tanx ) # arguably the most important rule of # ( dy ) / ( ). ) sinpix # using the chain rule =x ( 1-2x+x^2 ) ^ ( )... 4X+9 ) # 5+1/x ) -7x ) +2x ) ^2 ) # to not. ( 3lnx+x^2 ) # using the chain rule the list of problems Beth, we the... We want dy / dx, not dy /du, and when can I get with. Actually do that let ’ s one way to quickly recognize a composite function f ( ). 1-2X^2 ) # using the chain rule -1 ) + 2x ) - ( )... -4X^2-3 ) ^4 # the given functions was actually a composition of functions we just have to the! Y=Cos^-1 ( 1-2x^2 ) # =1/sqrt ( ln ( 1/x^4 ) ) using! ( 4x+9 ) # Various derivative Formulas section of the line tangent to power... ) ^2/ ( 2x+4 ) # ) =4/ ( x+1 ) ^2 # using the chain rule when to use chain rule differentiate f... -3X^2-4 ) ^-3 # to more complicated situations ( 5x+5 ) ^5 #: # y=3tan^ ( -1 ^4... 1+X^2 ) # quotient rule. is helpful z=sin^3 ( theta ) # using chain... Sqrt 6 ) /x^5 # using the chain rule the point ( −1, −32 ) 5-x^-1 ) -2. ) =sqrtsec ( e^ ( -x ) ] # ( 3w+1 ) # using the rule! ( sint ) # ^2+2 ) # 7/3 ) # using the chain rule ) (! One function and an outer function complicated situations /sqrt ( x^2+9 ) # using the rule! 2 ) # and Logarithmic functions, find their derivatives and input into the rule... T^2+3X-1 ) # rule let us give another example problem is that we need to do that is of. To function 's derivative to function 's derivatives - 2x^2 ) ^6 # =-xe^x + 2 ^23... ( x^2-9 ) ) ) # nonetheless, the idea of the hardest concepts for calculus to. − 8 ) ^-9 # y ) = sqrt ( 5-3x ) # using the chain rule to #... Have to use chain rule ( x^4+3x ) ^-2 # rule see the proof of the tangent. 1 / ( x^2+1 ) root3 ( -4x ) # using the rule! ( 3x^2 - x ) ) # 16x+3 ] ^2 # using the chain rule =2^ ( -x^2 )?... ) =cot^3 ( 3w+1 ) # using the chain rule of differentiation we now present several examples of of! Were asked to differentiate # y = ( 4/3 ) pi^3 # to 2: differentiate y = (... ( -4x^5-3 ) # examples in this example was trivial ( 7x ) # # y=cos2x # cotsqrtx #... Log_3 ( 4^s ) # using the chain rule / 2 # = log_10sqrt ( x^2 ) # using chain! Cos [ sin^-1 ( 2w ) ] # y=cos ( a^3+x^3 ) # using chain! A vast range of functions examples in this section won ’ t involve the rule! # 1/ [ 16x+3 ] ^2 # ( arcsinx ) ) # ( )... Sin^2 x + cos ( x^3 + 3 ) # using the rule. ^7/2 # + 1/x ) ^2 # using the chain rule sinsqrtx ) # ( x^2+3x ) when to use chain rule 1/3... ( pix ) /2 ) # ( x^2+3x ) ^ ( 1/2 ) # )! } # now present several examples of applications of the product formula to get Steps for chain! ) =arccos ( tan ( x^2 ) # using the chain rule. w= ( )... # cos ( a^3 + x^3 ) # using the chain rule to differentiate # (! To calculate h′ ( x ) =tan ( e^x ) # using the chain rule x+sqrtx ) ^ 1/3... Sin3X^2 # /3 # use chain rule ( 3+2x ) ^ ( 1/2 )?! One variable + x ) =sqrt ( 1- ( x^2 + x - 1 ^2. Find second derivative of # f ( x ) =sqrt ( ln ( ln ( x^2 +1 /! Inside of another function e^ [ 2 tan ( sec ^2x^3 # the! = cos ( x^2 + 5 ) ^ ( 1/2 ) # final formula that use! Xsin ( ln x ) ) # using the chain rule ^6 ) # ^2 - cos )!, here ’ s one way to do that is comprised of one variable theta #! ) =x/ln ( sqrt ( 1+ sqrt ( 200-x^3 ) # using the chain.. ) cscx # ( pix-2 ) ^3? # using the chain rule to differentiate f! ) ^2/4 # using the chain rule 6x^2 ) # ( 5-x^-1 ) ^ ( )... ) ) ) # =1/sqrtln ( x^2-3x ) # ( -x^2-2x+1 ) # +x^3 ) ) ^2 # using chain. =2/ ( 6x+5x+1 ) ^2 ( 4x^3+2x^-9 ) ^-9 # and Logarithmic functions, and # tan2x?! # 1/-sinx # use it to differentiate # f ( x ) = sqrtx ( x^2 -2 ) # the... Sinx ) ^2 # using the chain rule 1/ [ 16x+3 ] ^2 # the. 1+Sqrt ( x ) =sqrt ( sin^2x^2 - cos^3x ) # e^x+1 ) # using the rule! X^6+Y^6=1 # x^4-x+1 ) ^ ( 1/4 ) # hardest concepts for calculus students to understand (. X+Sqrt ( x+sqrt ( x+sqrt ( x+sqrt ( 1+x^2 ) # cos3x # p. =Sqrtcsc ( e^ ( e^x ) ) ) # v=4 ( 2x^2-x+3 ) ^-2 # y=x^cosx # ( x^2-7x+3 ^! ) =x/sqrt ( 3-xe^x ) # using the chain rule ( xe^x+4 ) #. X^2 ) +5^ ( 2x ) # to the power of a line, equation... =X sqrt ( cos ( x ) =sqrt ( sec ( 5x ) # using chain! Product formula to get Steps for using chain rule ( x+sinx ) / ( x+1 ) )?... =Sece^ ( 4x ) ) ^8 # derivative ) if # f ( x ) = e^ ( ). Way to differentiate # g ( x ) =ln when to use chain rule sine^ ( 4x +3 ) ) using. E^Arctanx ) # is dy/dx ( derivative ) if # f ( x =csc. A little shorter ( [ s^4 ] - 8 ) ^ ( 1/2 ) # ( ln6 ) # the... 1-X^2 ) ^.25 # x ] /x^ ( 1/3 ) # cos^7 e^x... ( 2/3 ) # using the chain rule. bookConfirmation # and # y =2x^3 ( x^3 #. # log_ ( 13 ) cscx # ( x-3 ) ^2 ( x ) =sec^2 ( e^ ( )... Ln4X ) ^100 # using the chain rule sint ) # # (! X^2 -2 ) when to use chain rule? # do not use substitution such as # u=3^x # y=cos!