The statement "The derivative of an even function is always an odd function" is Option A: TRUE.

To understand why this is true, let's first define even and odd functions:

An even function is a function $ f(x) $ that satisfies the condition:

$$ f(-x) = f(x) $$

An odd function is a function $ g(x) $ that satisfies the condition:

$$ g(-x) = -g(x) $$

Now, let's consider the derivative of an even function. Let $ f(x) $ be an even function, meaning:

$$ f(-x) = f(x) $$

We want to find the derivative $ f'(x) $ and determine if it is odd. Using the definition of the derivative, we have:

$$ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} $$

Since $ f(x) $ is even, we have:

$$ f(x) = f(-x) $$

Taking the derivative of both sides with respect to $ x $, we get:

$$ f'(-x) \cdot (-1) = f'(x) $$

or

$$ f'(-x) = -f'(x) $$

This shows that the derivative $ f'(x) $ satisfies the condition for being an odd function. Therefore, the derivative of an even function is indeed always an odd function.

JEE Advanced Subjects