Question

Differentiate: y= X^(e^x)

Answer 1

the derivate of e^x is e^x - you agree?
So use the power rule

Answer 2

you must take the natural log of both sides

Answer 3

then you must apply log rules and then solve for the y'

Answer 4

Since \(x\) is a variable, you do not want to use the power rule, as that's only used for constant exponents. However, you will need to put everything inside a natural log. So you get\[\ln(y)=\ln(x^{e^x})=e^x\ln(x)\]Now you can differentiate both sides using the product rule and implicit differentiation (for the function \(y\)).

Answer 5

Once you've solved for \(dy/dx\) in terms of \(x\) and \(y\), you can plug the equation \(y=x^{e^x}\) in to get \(dy/dx\) purely in terms of \(x\).

Answer 6

Have you arrived at an answer yet? :)