Question

How do you differentiate \[\cos (-x)\]?

Answer 1

Hint: 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.