Partial derivative of composite function

Question

Given, \begin{align} f(x, y, z) & = x^{2} \times y \times 2z \tag{1}\\ g(f(x, y, z), a) & = f(x, y, z) \times a \tag{2}\\ h(g(f(x, y, z),a)) & = g(f(x, y, z), a) - 1 \tag{3} \end{align}

How to find $\dfrac{\partial h(g(f(x, y, z),a)) }{\partial x}$?

My answer: \begin{align} \dfrac{\partial h(g(f(x, y, z),a)) }{\partial x} & = a \times 2xy \times 2z \end{align}

Would you be so nice and let me know if I got the correct answer?

Answer 1

You have

$$h(g(f(x, y, z),a))=2ax^2yz-1$$

and indeed

$$\frac{\partial h(g(f(x, y, z),a))}{\partial x}=4axyz.$$