Question

What is the derivative of `f(x)=cos(-x)-cos(x)`?

Answer 1

It's just a tricky way to write the identitcal-zero function, so the derivative is zero.

Explanation:

I'd say that no derivatives are needed, since you know that `cos(-x)=cos(x)`, and so

`cos(-x)=cos(x) = cos(x)-cos(x) = 0`, and the derivative of `0` is of course `0`.

Anyway, even if we didn't notice that, we can do the derivatives: since the derivative of `cos(x)` is `-sin(x)`, we have (using the chain rule for the first term)

`-sin(-x) * (d/dx (-x)) - (-sin(x))`

and since `(d/dx (-x))=-1`, the expression becomes

`sin(-x) +sin(x)`

Again, you should use the fact that `sin(-x)=-sin(x)`, and so the sum is again zero.