product rule constants

Question

I'm struggling with the concept of constants in the context of derivatives. For example;

$$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant. $$f'(\theta)=-r\sin\theta$$

Why is the product rule not used in the above?

Answer 1

You can still apply the product rule in the following way: $$ f'(x)= \frac {dr}{dx}(cos\theta-1)+r \frac{d(cos\theta-1)}{dx} = 0(cos\theta-1)+r(-sin\theta+0)=-rsin\theta $$

Since we have the following differentiation rule (which is a direct corollary of the product rule):

$$ \frac{d(kf(x))}{dx} = k\frac{df(x)}{dx}$$ where $k$ is a constant,

we usually do not use the product rule in this situation.

Answer 2

You can use the product rule (if you like). Just need to note that the derivative of a constant ($r$ here) is $0$, and you will get the same result.