(A) By definition,a function $f(x)$ is even if $f(-x) = f(x)$.Differentiating both sides with respect to $x$ using the chain rule,we get:$\frac{d}{dx}

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(A) By definition,a function $f(x)$ is even if $f(-x) = f(x)$.Differentiating both sides with respect to $x$ using the chain rule,we get:$\frac{d}{dx}[f(-x)] = \frac{d}{dx}[f(x)]$$f^{\prime}(-x) \cdot (-1) = f^{\prime}(x)$$-f^{\prime}(-x) = f^{\prime}(x)$$f^{\prime}(-x) = -f^{\prime}(x)$Since $f^{\prime}(-x) = -f^{\prime}(x)$,the derivative $f^{\prime}(x)$ is an odd function.

Class 12 Mathematics Continuity and Differentiation Derivative at a point, Standard differentiation

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If $f(x)$ is an even function,then $f^{\prime}(x)$ is

KCET 2012 All KCET

A
an odd function
B
an even function
C
may be even or may be odd
D
nothing can be said

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