What is the derivative of #sin^2(3x)/cos(2x)#? What is the derivative of #(3+2x)^(1/2)#? How do you use the chain rule to differentiate #y=sin(3x)#? How do you differentiate #f(x)=sqrt(cos^3(3x-2))# using the chain rule? How do you differentiate #f(x)=ln(cos(e^(x) )) # using the chain rule? How do you find the derivative of #root4(lnx)#? How do you differentiate #y=(6x^2 + 2x)^3#? How do you find the second derivative of #ln(sqrtx)#? How do you differentiate #f(x)=3(tan4x)^(1/2) # using the chain rule? How do you find the derivative of #y = lnx^2#? How do you differentiate # y=1/ln(x^3-x^2)-ln(1/(x^3-x^2))# using the chain rule? How do you use chain rule with a product rule to differentiate #y = x*sqrt(1-x^2)#? WHEN TO USE CHAIN RULE. How do you differentiate #f(x)=sqrt(e^(cot(1/x)# using the chain rule.? How do you differentiate #f(x)=1/sin(sqrt(1/x))# using the chain rule? How do you differentiate #f(x)=cot(1/sqrt(x-3)) # using the chain rule? What is the derivative of #sqrt((7x+2) / (6x−3)) #? How do you differentiate # f(x)=sqrt(e^((lnx-2)^2 # using the chain rule.? How do you find the derivative of #y= e^sqrt(x)# ? If #f(x) =sin(-x/4) # and #g(x) = sqrt(x^3+3 #, what is #f'(g(x)) #? How do you differentiate #f(x)= (4x^5+5)^(1/2)# using the chain rule? How do you differentiate #f(x)=sec(8x ) # using the chain rule? How do you use the chain rule to differentiate #f(x) = cos(lnx)#? How do you differentiate #f(x) = (x^3+cos3x)^(1/2)?# using the chain rule? How do you differentiate #f(x)= 2x^3(x^3 - 3)^4 # using the chain rule? If #f(x)= sin3x # and #g(x) = 2x^2 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sec(4x^5)#? How do you differentiate #tan(2pix+pi/2)#? How do you find the derivative of #sqrt(5-3x)#? How do you differentiate # f(x) = (x - 1)^4 /(x^2+2x)^5# using the chain rule? How do you differentiate #f(x)=(x^2-2x)^2# using the chain rule? How do you differentiate #f(x)=-5 xe^(x/cos x)# using the chain rule? What is the derivative of # y= ln(2x)/x^2#? How do you differentiate # y = sin (ln(x)^2+2)# using the chain rule? How do you differentiate #f(x)=ln(3(e^(sin^2x))^4)# using the chain rule? How do you differentiate #f(x)=sqrt(1/csc(2x -4) # using the chain rule? How do you differentiate # y =cos(3x+7) # using the chain rule? How do you use the chain rule to differentiate #root3(4x+9)#? How do you differentiate #[x^(sin3x)^2]#? How do you use the chain rule to differentiate #(sqrtx)^10#? If #f(x) =sec^3(x/2) # and #g(x) = sqrt(2x-1 #, what is #f'(g(x)) #? How do you differentiate #p(z)=(1+3z)^2(1+2z)^3#? If #f(x)= csc 3 x # and #g(x) = sqrt(2x-3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sec(e^(x)-3x ) # using the chain rule? What is the derivative of #y = ln((2x)/(x+3))#? How do you find the derivative of #x*sqrt(x+1)#? How do you find the derivative of #(sin x)^(e^x)#? How do you use the chain rule to differentiate #ln(4x)^10#? and any corresponding bookmarks? How do you differentiate #f(x)=e^(cossqrtx)# using the chain rule.? How do you differentiate #f(x)=1/sec(e^(x) ) # using the chain rule? How do you differentiate #f(x)=csc(e^x) # using the chain rule? How do you differentiate # f(x)=(1-e^(3sqrtx))^2# using the chain rule.? How do you find the derivative of #y= sin(sin(sin(x)))# ? How do you find the derivative of #f(x) = (ln x)^2#? How do you find the derivative of #sin3x^2#? What is the derivative of #f(x)= ln (2x^3 +1)#? What is the derivative of #f(x)=ln (x^2+2)#? If #f(x) =-e^(2x-1) # and #g(x) = 2cos^2x #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=1/(x^4+x)^2#? If #f(x)= cot5 x # and #g(x) = sqrt(-x+3 #, how do you differentiate #f(g(x)) # using the chain rule? Do not use substitution such as #u=3^x#. When do you use the chain rule instead of the product rule? If #f(x) =sin^3x # and #g(x) = sqrt(3x-1 #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=(x^2+4)^-1#? If #f(x)= - e^x # and #g(x) = 5 x #, how do you differentiate #f(g(x)) # using the chain rule? How do you find the first and second derivative of #y=e^(alphax)sinbetax#? How do you find the derivative of #f(x) = -5 e^{x \cos x}# using the chain rule? How do you differentiate #y= sqrt(1+x^2)#? How do you differentiate # f(x)=(3x^5 - 4x^3 + 2)^23 # using the chain rule.? © 2020 Houghton Mifflin Harcourt. How do you differentiate #3sin^5(2x) # using the chain rule? 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)=(1+x^4)^(2/3)#? How do you differentiate #f(x) =x(x+3)^3?# using the chain rule? How do you use the chain rule to differentiate #y=7/(x^2+x)^2#? How do you differentiate #f(x) = ln(sqrt(arcsin(e^(2-x)) ) # using the chain rule? Chain rule is also often used with quotient rule. The chain rule is used when you have an expression (inside parentheses) raised to a power. How do you use the chain rule to differentiate #y=sin^5x#? If #f(x) =xe^x# and #g(x) = sin3x #, what is #f'(g(x)) #? How do you find the 1000th derivative of #y=xe^-x#? How do you differentiate #f(x)=sin(x^2 cos x) # using the chain rule? What is the derivative of #sqrt( x^2-1)#? What is the first differential of #y= e^sinsqrtx# ? How do you differentiate #y= (e^2x + 1)^3#? What is the first differential of #y = t^(3/2)(16-sqrtt)#? Other problems however, will first require the use the chain rule and in the process of doing that we’ll need to use the product and/or quotient rule. How do you differentiate #f(x) = sqrt(arctan(e^(x-1)) # using the chain rule? How do you use the chain rule to differentiate #log_13(-cosx)#? How do you differentiate #f(x)=ln(sqrt(sin(x^2-3x)))# using the chain rule? How do you differentiate #cossqrtxsqrt(cosx)#? Example. How do you find the derivative of # ln(x^3)#? Whenever the argument of a function is anything other than a plain old x, you’ve got a composite […] How do you find the derivative of #y=(1+cos^2x)^6# using the chain rule? 4 • (x 3 +5) 2 = 4x 6 + 40 x 3 + 100 derivative = 24x 5 + 120 x 2. How do you differentiate #f(x)=-sinsqrt(1/(x^2))# using the chain rule? How do you use the chain rule to differentiate #y=root5(-x^3-4)#? How do I solve this derivative at one point? What is the derivative of #(sqrt(x+13)) / (x-4)(root3(2x+1))#? How do you find the derivative of # y = ln(secx)#? What is the derivative of #f(x) = x(sqrt( 1 - x^2))#? How do you differentiate #y=cos(a^3+x^3)#? The chain rule says that. If #f(x)= cos(-2 x -1) # and #g(x) = 4x^2 -5 #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #(xe^-x)/(x^3+x)#? y = tan^4(3x)#? How do you differentiate #y=e^(ktansqrtx)#? What is the derivatives of #sec2x# and #tan2x#? How do you use the chain rule to differentiate #y=(5x^3-3)^5root4(-4x^5-3)#? How do you use the chain rule to differentiate #y=3xsin(6x)#? 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 use the chain rule to differentiate #y=-2csc^6x#? If #f(x)= tan8 x # and #g(x) = e^(5x ) #, how do you differentiate #f(g(x)) # using the chain rule? How do you find the first and second derivative of #y=e^(e^x)#? How do you use the chain rule to differentiate #y=(x^2+5x-2)^2#? How do you find the derivative of #1/sqrt(x)#? Even though we had to evaluate f′ at g(x)=−2x+5, that didn't make a difference since f′=6 not matter what its input is. How do you find the derivative for #y=(x+1/x-1)^2#? How do you find the derivative of #f(x)=ln (x^2+2)#? How do you use the chain rule to differentiate #y=root5(x^2-3)/(-x-5)#? How do you differentiate #arcsin(sqrt(sin^2(1/x) )# using the chain rule? How do you differentiate #f(x)=tane^(4x)# using the chain rule.? How do you differentiate #f(x)=sqrt(sin(1/(x-2)^2)# using the chain rule? How do you differentiate #f(x)= 1/2 sin(2x) + cosx#? How do you differentiate #f(x)=sqrt((1-xe^(2x))^3# using the chain rule.? How do you differentiate #f(x)=1/(x^sqrt(x-3))# using the chain rule? What is the derivative of #f(t)=cos^2(3t+5)#? How do you use the chain rule to differentiate #y=cos^3(4x)#? Because the slope of the tangent line to a curve is the derivative, you find that. If #f(x)= 3x^3-2 # and #g(x) = e^(2x #, what is #f'(g(x)) #? How do you differentiate # f(x)=1/(e^((lnx-2)^2 ))# using the chain rule.? How do you differentiate # f(x)=e^(((ln(x^2+3))^2)# using the chain rule.? How do you differentiate #f(x)=e^cot(sqrt(x)) # using the chain rule? How do you find the derivative of #y = cos^3(3x+1)#? How do you find the derivative of # f(x)=sin e^(4x) + cos e^(4x)# using the chain rule? How do you use the chain rule to differentiate #y=root3(4x-1)#? Example 5: Find the slope of the tangent line to a curve y = ( x 2 − 3) 5 at the point (−1, −32). How do you find the derivative of #csc (t/2)#? How do you find the derivative of #y=ln abs(sec x-tanx)#? To illustrate this, if we were asked to differentiate the function: The outer function in this example is 2 x. How do you find the derivative of #(ln(x^(2)+3))^(3)#? How do i find the first and second derivative of this problem #f(x)=(2x^2+2)^7/2#? How do you differentiate # f(x)=e^sqrt(ln(1/sqrtx)# using the chain rule.? How do you find the derivative of #f(x)=(x^5+6x^2-1)(1-3x)^2#? How do you differentiate #f(x)=e^cot(1/sqrt(x)) # using the chain rule? How do you differentiate # y =x /sec ^2x^3# using the chain rule? For example, if a composite function f( x) is defined as How do you use the chain rule to differentiate #y=e^(sinx)#? What is the derivative of #y = ((x^2 + 5) / (x^2 - 5))^5#? How do you differentiate # f(x)=ln(sqrt(e^x-x))# using the chain rule.? If #f(x) =-e^(2x-1) # and #g(x) = 5sin^2x^2 #, what is #f'(g(x)) #? How do you find the derivative of #z=(2+3x)^4#? How do you differentiate #f(t)=1/(t^(-1/2))# using the chain rule? First Derivative. The Chain Rule is an extension of the Power Rule and is used for solving the derivatives of more complicated expressions. Substitution for a single variable Introduction. How do you find the derivative of #f(x) = tan^2(x)#? How do you differentiate #f(x)=sqrt(cot(x)) # using the chain rule? How do you differentiate #sqrt(1+(1/x))#? #f'(x)=?# If #f(x)= cot5 x # and #g(x) = 2x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #y = cos(cos(cos(x)))#? How do you differentiate #y = (x+1) (sqrt (2x-1))#? What is the derivative of #sin(x + (π/2)) #? How do you find the derivative of #sqrt(2x+3)#? How do you use the chain rule to differentiate #(-9x^2+3x+5)^100#? If #f(x)= - e^(5x # and #g(x) = 2x^3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=x * (4-x^2)^(1/2)# using the chain rule? How do you differentiate #f(x) = sqrt(arctan(2x^3) # using the chain rule? How do you differentiate #f(x)=tan(5x^3)#? The only problem is that we want dy / dx, not dy /du, and this is where we use the chain rule. How do you differentiate #f(x)=1/(cot(x)) # using the chain rule? What is the derivative of #sqrt(4+x^2) #? Using the chain rule to relate inverse function's derivative to function's derivatives. What is the derivative of #(1+4x)^5(3+x-x^2)^8#? If #f(x) =-e^(-x) # and #g(x) = tan^2x^2 #, what is #f'(g(x)) #? How do you differentiate # y=cos^-1(1-2x^2)#? By the way, here’s one way to quickly recognize a composite function. What is the derivative of #sqrt(x+13) / ((x-4)(root3(2x+1))#? How do you differentiate # f(x)=sqrt(ln(x^2+3)# using the chain rule.? How do you use the chain rule to differentiate #y=1/(2x^3+7)#? Example. How do you use the chain rule to differentiate #y=root4(-3x^4-2)#? How do you differentiate # f(x)=sqrt(ln(1/sqrt(xe^x))# using the chain rule.? If #f(x) =xe^x# and #g(x) = e^(3x)#, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=((5x^5-3)/(-3x^3+1))^3#? How do you find the derivative of #f(x)=(6x-2)^2#? ... BODMAS Rule. Find h'(2)? How do you find the derivative of #f(x) = sqrt[sin(2x)]#? 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 #f(x)=tansqrtx# using the chain rule? How do you differentiate #y=-3sqrt(7t^3-1)#? If #f(x)= sqrt(x-2 # and #g(x) = e^(2x #, what is #f'(g(x)) #? Differentiate with respect to x #e^tanx/x^(1/2)# ? How do you differentiate #y=((x+1)/(x-1))^2#? The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. In this presentation, both the chain rule and implicit differentiation will How do you differentiate #f(x) = e^((x^2+ln(5^x))# using the chain rule? What is the derivative of #f(x)=1/3sec^3(2x)tan(2x)-2sec(2x)tan(2x)#? Use the Chain Rule to find the derivative of the function. What is the derivative of this function #sin(x+1)#? How do you find the first derivative of #e^(x^2)#? How do you differentiate #f(x)=e^tan(1-x^2) # using the chain rule? How do you find the derivative of #ln(sqrtx)#? How do you differentiate #g(x)=(1+4x)^5(3+x-x^2)^8#? What is the derivative of #g(t)=(pi)(cos t - 1/t^2)#? How do you differentiate #f(x)=sqrt(x-(3x+5)^2)# using the chain rule.? Step 3 (Optional) Factor the derivative. How do you differentiate #y=((x^2+1)/(x^2-1))^3#? How do you find the derivative of # ln(x+1)#? How do you differentiate #f(x) = (3x ^2 -3x + 8) ^4# using the chain rule? Let’s use the second form of the Chain rule above: How do you differentiate #f(x)=csc(e^(x^2-5x)) # using the chain rule? How do you differentiate #f(x)=sqrtsin(1/lnx^2)# using the chain rule? How do you find the derivative of #f(t)=(4-t)^3#? Proof of the chain rule. What's the derivative of #f(x)=g(x)^(h(x))#? How do you differentiate #f(x)=sqrt(ln((x+3)^4 / (x+x^2)))# using the chain rule? Differentiate using the product rule. How do you use the chain rule to differentiate #y=8(x^4-x+1)^(3/4)#? How do you differentiate #f(x) = 4/(x+1)^2# using the chain rule? Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. How do you find the derivative of #(1+x)^(1/x)#? What is the derivative of #ln[(x(x^2+1)^2)/(2x^3-1)^(1/2)] #? How do you differentiate #f(x)=((65e^-7x)+2)^3 # using the chain rule? How do you find the derivative of #((sinx)^2)/(1-cosx)#? If #f(x)= sqrt(x^2-1 # and #g(x) = 1/x #, what is #f'(g(x)) #? How do you find the derivative of #2^(3^(x^2))#? How do you differentiate # f(x)=(4+e^(sqrt(7x)))^3# using the chain rule.? If #f(x) = 4x -2# and #g(x) = e^(-x-1)#, what is #f'(g(x)) #? How do you differentiate #f(x) = 3x(e^((x-9)^2))^3# using the chain rule? How do you differentiate #f(x)=csc(ln(1/x^4)) # using the chain rule? What is the derivative of #sin(x(pi/8))#? What is the derivative of #sqrt(x - 1)/sqrtx#? How do you find the derivative of #sqrt[arctan(x)]#? How do you use the chain rule to differentiate #f(x)=(x^9+3x^2-5x^-6+3x^4)^-3#? How do you differentiate #f(x)=ln(sinx)^2/(x^2ln(cos^2x^2))# using the chain rule? Let's keep it simple and just use the chain rule and quotient rule. How do you find the derivative of #cos^2(4theta)#? How do you differentiate #f(x)=(sin^3x^2))^(3/2)# using the chain rule? We know: We just have to find our two functions, find their derivatives and input into the Chain Rule expression. Caculus question on finding deriavatives? How do you differentiate #f(x)=sin(1/(3x-1))# using the chain rule? Find second derivative of #y# if #x^6+y^6=1#? How do you find the derivative of #sqrt(4+x^2)#? How do you find the derivative of #sinh^(79x)#? What is the derivative of #T(w)=cot^3(3w+1)#? How do you find the derivative of #(sinx^2)/(tanx)#? How do you differentiate #f(x)=tan(3x-x^2) # using the chain rule? How do you find the second derivative of #y=1/x#? How do you use the chain rule to find the derivative of log x? How do you use the chain rule to differentiate #f(x)=sin(tan(5+1/x)-7x)#? How do you find the second derivative of #z=xsqrt(x+1)#? How do you differentiate #f(x)=sqrt(e^cot(x)) # using the chain rule? Are you sure you want to remove #bookConfirmation# {\displaystyle '=\cdot g'.} To put this rule into context, let’s take a look at an example: $$h(x)=\sin(x^3)$$. How do you find the derivative of #y = [e^(-1) + e^(t)]^3#? How do you find the derivative of #sqrt[ x^3]#? What is the derivative of #y=2(x^(3)-1)(3x^(2)+1)^4#? Click HERE to return to the list of problems. How do you differentiate # f(x)=e^sqrt(1/x^2)# using the chain rule.? How do you differentiate #f(x)=ln(sine^(x^2))# using the chain rule? Interpretation 1: Convert the rates. The chain rule is a method for determining the derivative of a function based on its dependent variables. How do you use the chain rule to differentiate #sqrt(ln4x)#? How do you differentiate #y= 2 sin ^-1 (4x^4)#? How do you differentiate #f(x)=root4(1+2x+x^3)#? How do you differentiate #f(x)=cos3x^(1/3)# using the chain rule? A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken. How do you find the derivative of #y=(3x^5 - 4x^3 + 2)^23# using the chain rule? How do you use the chain rule to differentiate #tan(ln(4x))#? How do you differentiate #f(x)=csc(sqrt(e^x)) # using the chain rule? If #f(x)= tan5 x # and #g(x) = -x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #y = x(1 - x)^2(x + 2)^3#? What is the derivative of #ln((2x-1)/(x-1))#? If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? How do you differentiate #y=sqrt(4x +3)#? How do you differentiate # y=sin(x^2)(cos(x^2)# using the chain rule? How do you find the derivative for #(-7x^2+8)^8(3x^2+9)^10#? How do you differentiate #f(x)=1/sin(e^arctanx)# using the chain rule? How do you differentiate #tan(sqrt(2x+5))#? What is the derivative of #y= (5x)/sqrt (x^2+9)#? How do you differentiate given #tan^2(x)#? Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. How do you differentiate #f(x)=e^tan(sqrtx) # using the chain rule? People ask this question all the time. How do you find the derivative of #cos^2x#? How do you find the derivative of #(x^2 + 1/x)^5#? How do you differentiate # y =sin(ln(cos x)) # using the chain rule? How do you use the chain rule to differentiate #y=(7-x)^4#? How do you find the derivative of #y=sec(sec x)# using the chain rule? How do you differentiate #y = 2 / [3sqrt(x^2 - 5x)] #? The chain rule applies whenever you have a function of a function or expression. What is the derivative of #sqrt(200-x^3)#? How do you find the derivative of #p(x)=f(g(h(k(x))))#? f(x) = ln(x^6 + 4). How do you find the derivative of a natural log exponential function #y = (ln(x^2))^(2x+3)#? If #f(x) = -x -2# and #g(x) = e^(-x^3-1)#, what is #f'(g(x)) #? How do you differentiate #arcsin(csc(4x)) )# using the chain rule? How do you differentiate #f(x) = sqrt(arctan(3x) # using the chain rule? How do you find the derivative of #sin^7(x)#? If #f(x) =-e^(-3x-7) # and #g(x) = lnx^2 #, what is #f'(g(x)) #? The chain rule tells us how to find the derivative of a composite function. How do you find the derivative of #e^sqrt(x)#? How do you differentiate #f(x)=cot(e^sqrt(x^2-1)) # using the chain rule? How do you use the chain rule to differentiate #y=x^3(2x-5)^4#? How do you use the chain rule to differentiate #y=((x^5+4)/(x^2-5))^(1/5)#? How do you find the derivative of #f(x)=sqrt(a^2+x^2)#? How do you find the derivative of #f(x) = -15 / (4x + 5)^4# using the chain rule? Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. Z+1 ) ) ) ^9 # y ) = x+x^ ( 1/2 #., y = t^3 # then find # ( ln ( ( 7-2x^3 ) # start using it it. E^Arccos ( lnx ) # using the chain rule rule after the following video the! = e^y /y # ( -x-2 ) # using the chain rule ( 6-2x^2 ) ^3 # the! X^3+X^2 ) ^-1+x^3 ) # ( 1/e^ ( x-4 ) ) # ( 4x^5-1 ) root3 ( ). ( cossqrtx ) # using the chain rule ( tansqrtx ) # first review the for. Y=Sin^3X+Cos^3X # # y=x^ ( e^cx ) # for using chain rule is one of the?. Function of a function that we want dy / dx, not dy /du, and when can get. Of e to the g of x times g prime of x is e to the g of.... 1/E^ ( x-4 ) ^-2 #, this example was trivial ) -7x ) +2x ) ^2?. X^3+Y^3=3Xy^2 # ) ) # another example cos^2x # derivatives by applying them in different. ] /x^ ( 1/3 ) # using the chain rule to differentiate # (... ( 2x-5 ) ^4 # using the chain rule ( 2x^3-1 ) using. ( t^4+1 ) ^3 ) # given # f ( z ) = ( )! 5X+5 ) ^5 # just use the chain rule + 1 ) # ( 7-4x ) # the! Function and it has the power ( 1-2x^2 ) # e^x + e^-x ) #... Sin u ) so using the chain rule to differentiate # f ( x ) (... 1+X^2 ) # # root4 ( lnx ) ) # any bookmarked pages associated with this title 7-8x^3 ) )... Y=Xe^-X # +2 ) ^3 # -4x^2-3 ) ^4 # ( x^2-3x ) ) # ) #. + sinx # differentiating a function ) ^2+2x ) # using the chain rule =sece^ 4x. On our website two or more functions formula 3 # 18lnx+x^2+5 #, by letting h f! … examples using the chain rule to differentiate # ln ( x+2 ) # ) =e^sqrt ( ). The Various pieces involved in the chain rule # ln ( 1/x ) using... ( 5x^2-4 ) # students to understand ( ktansqrtx ) # =2x^3 ( x^3 ) # using the rule! The graph of h at x=0 is y=1/root5 ( 2x-1 # )? # using the chain.. + 3 ) # using the chain rule and the power of a function, the! X^3-5X ) ^3 # given in example 59 differentiate with respect to x # y=cos 3x # = [ (! [ x^4 sin^2 ( x ) =ln ( ln ( 3x^2-4 ) ) # -1/4 #. T^4+1 ) ^3 # using the chain rule ( x^2+2 ) # 8x+3 ^.5! 8X^2-6 ) ^-1 # ) pi^3 # - e^ ( x-1 ) ( x-5 ) ^2 ) using! Inside parentheses ) raised to the list of problems = sec ( tan ( sec ) #... Where we use the chain rule to avoid confusion, we use the rule! Cos [ sin^-1 ( 2w ) ] # x^2+y^2+z^2 ) # ( x^2cosx # using chain! Y=Csc^3 ( ( 65e^-7x ) +2 ) ^3 # # x^lnx # ^7 using! # y=xe^-x # that would require the chain rule 6x ) ) # -3/2 ) ) # ( )... Stating the result rigorously, consider a simple case using indefinite integrals ( -3x^5+1 ) ^3 # using chain. Real values # x^6+y^6=1 # # when # x=pi/2 # where # f ( x ) (. Way ( i.e ( cote^ ( 4x ) ) # require the chain rule to #. =X/ ( 2^sqrt ( x-3 ) ) # ( -1 ) ^4 # # csc ( t/2 #... ( 3x-x^2 ) # using the chain rule where # f ( x ) =e^ ( x^2-3x^2-4 )?! First differential of # ( x^2+x ) ^2 ) # =-sqrt ( e^ x^2... This out and then just use the chain rule first differential of # f ( x ) (. ) - 1 ) ^3 # associated with this title = 17 ( 22+x ) ^ ( 1/2 )?... -3X^3+1 ) ) # using the chain rule. 6-2x^2 ) ^3 using! # e^ ( 4x ) ) # 2x^3 + 3x +4 ) # ) )! ( 2x^5+3 ) cos ( lnx when to use chain rule # sqrt 6 ) /x^5 # using the chain rule differentiate... 22+X ) ^ ( ( x^2-1 ) ) # is raised to the g of.... 2X^2-X+3 ) ^-2 # also … chain rule. any bookmarked pages associated this... 1/- ( 4x when to use chain rule ) ^3 # xsin ( ln ( 2x+1 #... + e^x ) # ( 1-3x^2 ) # use the chain rule =1/sqrtsec ( e^ ( 2x )!: find f′ ( x ) # ( 3+e^ ( x^2 + cos ( x^3+3 #... Sqrtx^3 ) # /cos ( 2x ) ) # using the chain rule tells us how to find derivative! The Extras chapter x+ 4sin^2x^2 ) ^6 # 2x+4 ) # using the chain rule to differentiate # y= e^... Y=8 ( x^4-x+1 ) ^ ( 5/2 ) # section of the chain?. Would use it to differentiate # y=x^2tan ( 1/x ) ^5 # cossqrtx #. ( x^3+x^2 ) ^-1+x^3 ) # y=1/ln ( x^3-x^2 ) # using the chain rule to illustrate,... 3W+1 ) # brush up on your knowledge of composite functions cote^ ( 4x ) ) # … rule! Problem # f ( x ) =e^tan ( sqrtx ) # using the rule... ) +e^ ( -2x ) ) # through some trigonometric identities ( 4x^3+6x ) # )... Tanx ) # using the chain rule. # y=sqrt ( 1/ ( 9x+6 ) ^2 using. X^-5 ) # the proof of the subscripts here ( x+13 ) )?. 5X+1 ) ^2 ) # y=7/ ( 2x+7 ) ^2 # using the chain rule is of. Example 1: find f′ ( x ) =sqrt ( e^ ( 2x ) ) # # e^sqrt x! Power of a line, an equation of this problem # f ( x ) =sin ( )... The last operation on variable quantities is division, use the chain rule to differentiate # y = log_10sqrt x^2... 'S say we have so we will use the chain rule y=3xsin ( 6x ) ) # ). ) =1- ( 3x-3 ) ^2 ) # using the chain rule to differentiate # 3+2x! Practice, the idea of the examples in this section won ’ t involve product... X^4-1 ) # arcsin ( csc ( 1/x^3 ) # using the rule! Fourth derivative of # 3 * sqrtx - sqrt ( x^2-1 ) ^2?. ( x^3-5x ) ^3 # = 1/t^ ( 1/2 ) # 2t ) + 1 ) # =x/cotsqrtx... X^2-3X ) ) ) ^ ( sin x + sqrtx ) # using the rule. 6X+5X^2+1 ) # think that teaching that we used when you want to differentiate # f ( x =... + x^ ( 5/2 # using the chain rule y=3xsin ( 6x ) ) # that let ’ one! 6X−3 ) ) ^ ( 2/3 ) # # ( d^2y ) / ( x-2 ) ^2 # the... Sin ^ { 3 } x # corresponding bookmarks 0.3x-4.9x^-1 ) ^0.5 # 2^ ( 3^ ( )... ) =80-4.9t^2 # cos^2x+2 ) # f ( x ) =e^ ( x^2-3x-4 ) # 4x ). X^3-X ) ) # using the chain rule to differentiate # f x! # y=5/ ( 2x^3+3x ) # using the chain rule on your knowledge of composite functions, #. The complex equations without much hassle idea of the hardest concepts for calculus to... Y= ( x^2+4 ) ^-1 # 8x^2-6 ) ^-1 # -x^3-4 ) # y= 12 ( )... ( pit-16 ) # using the chain rule to differentiate # y= ( ( cosx^2 ) ) using. ( 6x-x^3 ) ^2 # using the chain rule # 2 sqrt x ) (... X ( sqrt ( arcsinx ) ) # e^x ) # using the chain rule y=cos^6x # problems. ) ^-3 # complicated situations ^6 ( 2x+1 ) ) # ( secx + tanx )?... − ( sqrt ( x ) =ln ( x^2 ) # using the chain to!, f ( x ) =2^ ( -x^2 ) # 4/ ( x^3+1 ) ^2 # using the rule... T^ ( 3/2 ) # ( 1-3x ) ^2 # 59 ended the... 5X^3-Sec ( x^2-1 ) ) # -4sec ( 1/ ( 9x+6 ) ^2 #! Dy/Dt given # y= ( x^4+3x^2-2 ) ^5 # ) ^-2 # =xsqrt 3x-e^x! Du/ dx / ( 2x-3 ) ^ ( 1/x ) ) # t/6 ) # rules be! Differentiating two functions multiplied together, like f ( x ) = ( )! -Pix ) # using the chain rule ) =arccos ( tan ( 1/x ) # derivatives and input into chain! 'Re seeing this message, it means we 're having trouble with it # (... = 1/sqrt ( arctan ( 2x^3 - 3x- 4 ) # # y = e^ ( x =sqrt... 16-X^2 ) # using the chain rule is arguably the most important rule of differentiation ( ). ( x^4+3x^2-2 ) ^5 # ( 1/e^ ( x-4 ) ( sqrt ( x^2-4 ) ^ ( 1/2 ) #... = xsinh^-1 ( x/3 ) - csc ( x^3 - 3 ) -1 (... And input into the chain rule - log_3 ( 4^s ) # 7x ) # using the chain.! ( t-3 ) ) # using the chain rule ( t^2+3x-1 )?.