Question 1200824
<pre>
g(x) is an even function if and only if g(-x) = g(x)
g(x) is an odd function if and only if g(-x) = -g(x)
</pre>a) If f(x) is an odd function then f'(x) is an even function<pre>

{{{f(-x)=-f(x)}}}

Take derivatives of both sides:

{{{"f'(-x)*(-1)"}}}{{{""=""}}}{{{"-f'(x)"}}}

{{{"-f'(-x)"}}}{{{""=""}}}{{{"-f'(x)"}}}

Divide both sides by -1

{{{"f'(-x)"}}}{{{""=""}}}{{{"f'(x)"}}}

So f'(x) is an even function.

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</pre>b) If f(x) is an even function then f'(x) is an odd function<pre>

{{{f(-x)=-f(-x)}}}

Take derivatives of both sides:

{{{"f'(-x)*(-1)"}}}{{{""=""}}}{{{"-f'(-x)*(-1)"}}}

{{{"-f'(-x)"}}}{{{""=""}}}{{{"f'(-x)"}}}

Divide both sides by -1

{{{"f'(-x)"}}}{{{""=""}}}{{{"-f'(-x)"}}}

So f'(x) is an odd function.

Edwin</pre>