Implicit derivative of $x^y$

Question

So I have a problem that I can't solve for a couple hours now. Can anyone help me understand how to do an implicit derivative of $x^y$.

Thank you very much

Answer 1

If you want $\frac d{dx}x^y$, you write it using an exponential and use the chain rule $$\begin {align}\frac d{dx}x^y&=\frac d{dx}e^{y \ln x}\\&=e^{y \ln x}\frac d{dx}(y \ln x)\\&=e^{y \ln x}\left(\frac {dy}{dx}\ln x+\frac yx\right)\\&=x^y\left(\frac {dy}{dx}\ln x+\frac yx\right)\end {align}$$

Answer 2

Let $f(x)=x^{y(x)}.$ We wish to find $f'(x).$ We employ the usual trick of logarithmic differentiation: \begin{align*} \ln(f(x))&=y(x)\ln(x) \\ \frac{f'(x)}{f(x)}&=y'(x)\ln(x)+\frac{y(x)}{x} \\ f'(x)&=x^{y(x)}\left(y'(x)\ln(x)+\frac{y(x)}{x}\right). \end{align*}