Chain Rule

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Differentiate x^(lnx) using implicit differentiation and the chain rule

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1 of 10 videos by Tiago Hands

Key Questions

  • We can do a number of things ..

    1) Use the chain rule and quotient rule

    2) Use the chain rule and the power rule after the following transformations.

    #y=((1+x)/(1-x))^3=((1+x)(1-x)^-1)^3=(1+x)^3(1-x)^-3#

    3) You could multiply out everything, which takes a bunch of time, and then just use the quotient rule.

    Let's keep it simple and just use the chain rule and quotient 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'=3((1+x)/(1-x))^2*(1-x+1+x)/(1-x)^2#

    #y'=3((1+x)/(1-x))^2*(1+1)/(1-x)^2#

    #y'=3((1+x)/(1-x))^2*(2)/(1-x)^2#

    #y'=6((1+x)/(1-x))^2*(1)/(1-x)^2#

    #y'=6((1+x)^2)/(1-x)^2*(1)/(1-x)^2#

    #y'=6((1+x)^2)/(1-x)^4#

  • By Chain Rule and Power Rule,

    #y'=1/4(x^2+3x+5)^{1/4-1}cdot(x^2+3x+5)'#

    #=1/4(x^2+3x+5)^{-3/4}(2x+3)#

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