Where do I have to use Chain Rule of differentiation?

Question

I have come across many examples of chain rule of differentiation while studying physics (eg. finding velocity of SHM,differentiating Kinetic Energy with respect to time etc.).But,I feel I lack the intuition as to where I should use chain rule.I am understanding the problems when I look at the solutions but I am not myself able to say where I should use chain rule and where not?What are conditions necessary for using chain rule.

Someone please guide me!!

Thanks for any help!!

Answer 1

Edit: Thanks for pointing out the mistake. I dont have enough reputation to comment. Here I take mass to be constant.

Let us take an example in physics:

$$\text{K.E} = \frac{1}{2}mv^2$$ Differentiate both sides with respect to time: $$\frac{d}{dt}(K.E)=\frac{d}{dt}(\frac{1}{2}mv^2)$$ Now you have velocity which is changing but you are trying to differentiate it with respect to $t$. So here we have to use the chain rule. So by chain rule $$m v\cdot \frac{dv}{dt}$$ Mathematically given a composition of function $f(g(x))$ and you want to find its derivative with respect to $x$, using chain rule:

$$\frac{d}{dx}(f(g(x))=\frac{d(f(g(x))}{d(g(x))}\cdot\frac{d(g(x))}{dx}$$

Answer 2

Actually, the derivative of $\frac{1}{2}mv^2$, with respect to time, t, is $\frac{1}{2}\frac{dm}{dt}v^2+ mv\frac{dv}{dt}$. Your formula is assuming that m does not vary with time but you did not say that.