Find the following partial derivatives for a nested function?

Question

For $g(t,x) = f(t , h(t,x))$ find the following partial derivatives: $\frac{\partial g}{\partial t}$, $\frac{\partial g}{\partial x}$, $\frac{\partial^2 g}{\partial x^2}$

Answer 1

Note: you haven't given much info on the domain and codomain of $f$ and $g$, so I'll assume they are both functions $\mathbb{R}^2 \to \mathbb{R}$.

Consider the chain rule: Let $f(a,b)$ be a function in two variables where $a = a(t,x)$ and $b = b(t,x)$, and take $g(t,x) = f(a(t,x),b(t,x))$. Then: $$\frac{\partial g}{\partial t}(t,x) = \frac{\partial f}{\partial a}(a(t,x),b(t,x))\frac{\partial a}{\partial t}(t,x) + \frac{\partial f}{\partial b}(a(t,x),b(t,x))\frac{\partial b}{\partial t}(t,x)$$ and a similar formula exists for the partial derivative to $x$.

Let's do the the first one: in your case, we have: $$\frac{\partial g}{\partial t}(t,x) = \frac{\partial f}{\partial a}(t,h(t,x))\frac{\partial a}{\partial t}(t,x) + \frac{\partial f}{\partial b}(t,h(t,x))\frac{\partial b}{\partial t}(t,x) $$

However, we know that $b(t,x) = h(t,x)$, and that $a(t,x) = t$, hence we obtain: $$\frac{\partial g}{\partial t}(t,x) = \frac{\partial f}{\partial a}(t,h(t,x)) + \frac{\partial f}{\partial b}(t,h(t,x))\frac{\partial h}{\partial t}(t,x) $$ Can you do the other derivatives yourself?