MHTCET 18 Apr 2026 Shift 2 Paper

Concept:The derivative of an even function is always an odd function.Explanation:For an even function, we have $f(-x) = f(x)$ for all $x \in \mathbb{R}$.Differentiate both sides with respect to $x$.Using the chain rule on the left side:$\frac{d}{dx} f(-x) = f'(-x) \cdot (-1) = -f'(-x)$And the derivative of the right side is:$\frac{d}{dx} f(x) = f'(x)$Equating both derivatives gives:$-f'(-x) = f'(x)$Rearranging, we get:$f'(-x) = -f'(x)$This is exactly the defining condition of an odd function.Therefore, wherever $f'(x)$ exists, it must be an odd function.Options A, B, and C are not necessarily true for every even function.Answer:Option D: $f'(x)$ is an odd function.