Question

Prove the derivative of an even function is an odd function using the definition of a derivative

Answer 1

\[f'(-x) = \lim_{h \rightarrow 0} [f(-x+h)-f(-x)]/ h\]

Answer 2

\[= \lim_{h \rightarrow 0} [-f(x-h)+f(x)]/h\]

And now I'm stuck.

Answer 3

\[\cdots = \lim_{h\rightarrow 0} \frac{f(x-h)+f(x)}{-h}\]So it's just what we had before except with \(-h\) instead of \(h\). Now note that \(h \rightarrow 0\) is the same as saying \(-h \rightarrow 0 \).

Answer 4

Slight correction:\[\cdots = \lim_{h \rightarrow 0} \frac{f(x-h)\color{red} - f(x)}{-h}\]