Question

If $f\left(x\right)$ is an even function, where $f\left(x\right)\ne 0$, then which one of the following is correct?

• A. $f^{\prime }\left(x\right)$ is an even function
• B. $f^{\prime }\left(x\right)$ is an odd function
• C. $f^{\prime }\left(x\right)$ may be an even or odd function depending on the type of function
• D. $f^{\prime }\left(x\right)$ is a constant function

Answer 1

The correct option is B $f^{\prime }\left(x\right)$ is an odd function

Let $f\left(x\right)$ be an even function

$\Rightarrow f\left(x\right)=f(-x)$

Differentiate both sides, we get
$\frac{df}{dx}=\frac{d\left(f\right(-x\left)\right)}{dx}...\left(1\right)$

To differentiate $f(-x)$, we use the chain rule formula as follows:
Let $u=-x$, hence $\frac{d\left(f\right(-x\left)\right)}{dx}=\frac{df\left(u\right)}{du}\cdot \frac{du}{dx}=f^{\prime }\left(u\right)\cdot (-1)=-f^{\prime }(-x)$

Substituting in $\left(1\right)$ we get

$f^{\prime }\left(x\right)=-f^{\prime }(-x)$

That is $f^{\prime }\left(x\right)$ is an odd function.