Partial Differentiation: Suppose $f(r,\theta,\phi)$ and $x=r\sin(\theta)\cos(\phi)$. How to find $∂f/∂x$?
I have the following question and no access to solutions.
The variables $x$, $y$, $z$ and $r$, $θ$, $φ$ are connected by the following equations:
\begin{align} x&=r\sin(θ)\cos(φ) \\ y&=r\sin(θ)\sin(φ) \\ z&=r\cos(θ) \end{align}
(a) Find $∂x/∂φ$, $∂y/∂φ$, and $∂z/∂φ$.
(b) Show that for any differentiable function $g(x,y,z)$ we have $g_φ(x,y,z)=x\cdot g_y(x,y,z)−y\cdot g_x(x,y,z)$.
(c) Suppose $f(r,θ,φ)=\cosθ$. Find $∂f/∂x$, giving your answer in terms of $x$, $y$, and $z$.
I am happy with parts a and b. Unfortunately, I am a little stuck with (c).
Any help or pointers would be much appreciated.
Thank-you for your help in advance.
Hint
Use that
$$f(r,\theta,\varphi)=\cos \theta=\dfrac{z}{r}=\dfrac{z}{\sqrt{x^2+y^2+z^2}}.$$