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Trigonometry
Question

Find the derivative of the trigonometric function....

Find the derivative of the trigonometric function. 

\( y = \sin ( ( \pi x ) ^ { 5 } ) \) 

\( y ^ { \prime } ( x ) = \) 

Answer

\(\frac { d } { d x } ( \sin ( ( \pi x ) ^ { 5 } ) ) = \cos ( \pi ^ { 5 } x ^ { 5 } ) \cdot 5 \pi ^ { 5 } x ^ { 4 } \) 

Steps 

\( \frac { d } { d x } ( \sin ( ( \pi x ) ^ { 5 } ) ) \) 

Simplify \( \sin ( ( \pi x ) ^ { 5 } ) : \sin ( \pi ^ { 5 } x ^ { 5 } ) \) 

\( = \frac { d } { d x } ( \sin ( \pi ^ { 5 } x ^ { 5 } ) )\) 

Apply the chain rule: \( \cos ( \pi ^ { 5 } x ^ { 5 } ) \frac { d } { d x } ( \pi ^ { 5 } x ^ { 5 } ) \) 

\( = \cos ( \pi ^ { 5 } x ^ { 5 } ) \frac { d } { d x } ( \pi ^ { 5 } x ^ { 5 } ) \) 

\( \frac { d } { d x } ( \pi ^ { 5 } x ^ { 5 } ) = 5 \pi ^ { 5 } x ^ { 4 } \) 

\( = \cos ( \pi ^ { 5 } x ^ { 5 } ) \cdot 5 \pi ^ { 5 } x ^ { 4 }\) 

Solution
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