First we apply the product rule.
f ′ ( x ) = ( x c ) ′ sin ( a x ) + x c ( sin ( a x ) ) ′ . {\displaystyle f'(x)=(x^{c})'\sin(ax)+x^{c}(\sin(ax))'.}
According to the power rule, the derivative of x c {\displaystyle x^{c}} is c x c − 1 {\displaystyle cx^{c-1}} .
For the derivative of sin ( a x ) {\displaystyle \sin(ax)} , we use the chain rule:
( sin ( a x ) ) ′ = cos ( a x ) ⋅ ( a x ) ′ = a cos ( a x ) {\displaystyle (\sin(ax))'=\cos(ax)\cdot (ax)'=a\cos(ax)} .
We get the final result: f ′ ( x ) = c x c − 1 sin ( a x ) + a x c cos ( a x ) {\displaystyle f'(x)=cx^{c-1}\sin(ax)+ax^{c}\cos(ax)} .
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, we use the chain rule:
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