Find the derivative of $y=x^{\sin x}$ Could someone please explain step 3 for the following: Why do they multiply $1/y$ with $y'$? I understand that the derivative of $\ln y$ is $1/y$, but I don't understand why it is multiplied with $y'$ in step 3. 
Find the derivative for    $y=x^{\sin x}$
Step 1: $\ln y=\ln x^{\sin x}$
Step 2: $\ln y=\sin x\ln x$
Step 3: $\frac{y'}y=\cos x\ln x+\frac{\sin x}x$
Step 4: $y'=y\left[\cos x\ln x+\frac{\sin x}x\right]=x^{\sin x} \left[\cos x\ln x+\frac{\sin x}x\right]$

The derivative of $\ln y$ with respect to $y$ is certainly $1/y$, but in this case the derivative needs to be taken with respect to x, so by the chain rule, the derivative of $\ln y$ with respect to $x$ is $y'/y$. 
