If f(x)=e^2x and g(x)=lnx, then determine the deriviative of y=f(g(x)) at x=e

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x)=e^2x and g(x)= ln x,

y = f(g(x))

=> y = `e^(2*ln x)`

y' = `e^(2*ln x)*2*(1/x)`

=> `e^(ln (x^2))*2*(1/x)`

=> `x^2*2*(1/x)`

=> 2*x

At x = e

y' = 2*e

The required value of `dy/dx` at `x = e` is `2*e`