Chain Rule - $z''(t) = z'(t) \frac{dz'(t)}{dz}$

Question

all. This question may be a trivial answer, but I'm having trouble figuring this out. I'm taking a course in classical mechanics and I don't exactly see how the following equation is a result of the chain rule. My professor keeps saying "obviously" when calling on it, but I do not understand how it follows.

For a function $z=z(t)$, my professor claims the following: $$ z''(t) = z'(t) \frac{dz'(t)}{dz} $$

How does this follow from the chain rule definition?

Thank you.

Answer 1

The chain rule says that $$\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}u}{\mathrm{d}v}\frac{\mathrm{d}v}{\mathrm{d}x}$$ So in your case: $$\frac{\mathrm{d}z'}{\mathrm{d}t}=\frac{\mathrm{d}z'}{\mathrm{d}z}\frac{\mathrm{d}z}{\mathrm{d}t}$$

Answer 2

$$z''(t) = z'(t) \frac{dz'(t)}{dz}=z'(t) \frac{dz'(t)}{dt}\frac{dt}{dz(t)}=\frac{z'(t)z''(t)}{z'(t)}$$