Using the chain rule for proof

Question

how do you use the chain rule to show that:

Thank you.

Answer 1

I'll show you how to find $\dfrac{dz}{dt}$ and leave you to find $\dfrac{d^2z}{dt^2}$.

We have $z = f(x, y) = f(g(t), h(t))$, so $z$ can be viewed as a function of $x$ and $y$, or as a function of $t$ alone. By the chain rule, we have

$$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = \frac{\partial f}{\partial x}\frac{dg}{dt} + \frac{\partial f}{\partial y}\frac{dh}{dt}.$$

In this case, $y = h(t) = c$ is constant, so $\dfrac{dh}{dt} = 0$. Therefore,

$$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dg}{dt}.$$