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Trigonometry
Question

Find the derivative of the trigonometric function.

$$y = \sin ( ( \pi x ) ^ { 5 } )$$

$$y ^ { \prime } ( x ) =$$

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

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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