Multi-variable calculus involving $\ln$

Question

I am having difficulty with differentiating this equation with respect to $y$: $$ W= x^{y \ln(z)}. $$ Differentiating calculators are giving me the answer

$$\ln(x) \ln(z).x^{y \ln(z)}$$

But I can't understand why the $\ln(x)$ appears in the answer

Any help would be greatly appreciated.

Answer 1

Start by taking the natural logarithm to both sides

$$ \ln W(x,y,z) = y\ln(z)\ln(x), $$

then differentiate w.r.t. $y$

$$ \frac{W_y}{W} = \ln(z)\ln(x) $$

which gives

$$ W_y = W \ln(z)\ln(x) \longrightarrow (1)$$

Now substitute $W = x^{y\ln(z)}$ in $(1)$ yields the desired result

$$ W_y = x^{y\ln(z)} \ln(z)\ln(x) $$