Question

How do you differentiate $\cos (-x)$?

Answer 1

We can see that the cosine function has a negative angle, but we know that cosine function is an even function, so we have $\cos (-x)=\cos x$. Now, we will differentiate the expression $\cos x$. And so we will straightaway write the derivative of the $\cos x$, hence we have the differentiation of the given function.

Complete step by step solution:
According to the given question, we have to find the differentiation of the given function.
The expression we have is $\cos (-x)$.
We can see that the cosine function in the given expression has a negative angle. But, we need not differentiate a negative angle. As we know that cosine function is an even function.
A function can be either an odd function or an even function, what it means is,
If a function is an odd function, then $f(-x)=-f(x)$ and
If a function is an even function, then $f(-x)=f(x)$.
For example –
sine function is an odd function, that is, $\sin (-x)=-\sin x$
cosine function is an even function, that is, $\cos (-x)=\cos x$

So, the expression we have is,
$\cos (-x)$
$\Rightarrow \cos x$
as the cosine function is an even function
Now, we will take the derivative of $\cos x$, we have,
$\dfrac{d}{dx}(\cos x)$
$\Rightarrow -\sin x$
Therefore, the differentiation of $\cos (-x)$ is $-\sin x$.

Note: It is advisable to know the functions which are odd and which are even, so that the expressions can be solved easily and faster too. While taking the derivative of a function, stepwise fashion should be implemented to prevent errors.