Question

If `f(x)= 3cos(2x) + sin(2x)`, how can I find `f'(x)`?

Answer 1

First of all, let's remark that the derivative of a sum is the sum of the derivatives, and so we can study the function one addend at the time.

In both cases, anyway, you'll need the chain rule, a formula which tells us how to derivate composite function.

By composite function, we mean a function which evaluates another function, as when we write `f(g(x))`. The derivative of such a function is `f'(g(x)) g'(x)`.

Studying the first addend, we have that `f(x)=\cos(x)`, and `g(x)=2x`.

Since the derivative of `\cos(x)` is `-\sin(x)`, and the derivative of `2x` is `2`, we have that
`d/dx 3\cos(2x) = 3 d/dx \cos(2x) = 3(-\sin(2x) \cdot 2)`

Similar steps lead us to say that
`d/dx \sin(2x) = \cos(2x)\cdot 2`. Adding the two, one has that

`d/dx 3cos(2x)+sin(2x) = 2(\cos(2x)-3\sin(2x))`