Question

How do you find the derivative of `x^tanx`?

Answer 1

`x^{tan(x)}(ln(x)*sec^{2}(x)+tan(x)/x)`

Explanation:

Use logarithmic differentiation: let `y=x^{tan(x)}` so that `ln(y)=ln(x^{tan(x)})=tan(x)ln(x)`.

Now differentiate both sides with respect to `x`, keeping in mind that `y` is a function of `x` and using the Chain Rule and Product Rule:

`1/y * dy/dx=sec^{2}(x)ln(x)+tan(x)/x`

Hence,

`dy/dx=y * (ln(x)sec^{2}(x)+tan(x)/x)`

`=x^{tan(x)} (ln(x)sec^{2}(x)+tan(x)/x)`