If $y = {sin}^{5} x$ $y=sin^5x$ then what is $\frac{d y}{d x}$ $dy/dx$ ?

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If $y = {sin}^{5} x$ $y=sin^5x$ then what is $\frac{d y}{d x}$ $dy/dx$ ?

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Answer 1 · 2016-10-04T23:53:42

Use the chain rule

Explanation:

If $y = {sin}^{5} x$ $y=sin^5x$ , then Let $u = sin x$ $u=sin x$ ,

Then $\frac{d u}{d x} = cos x$ $(du)/(dx)=cosx$ and $y = u^{5}$ $y=u^5$ so $\frac{d y}{d u} = 5 u^{4}$ $(dy)/(du)=5u^4$

Chain rule gives $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du) (du)/(dx)$ . so:
$\frac{d y}{d x} = 5 u^{4} cos x = 5 {sin}^{4} x cos x$ $(dy)/(dx)= 5u^4 cosx = 5sin^4x cos x$

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If $y = {sin}^{5} x$ $y=sin^5x$ then what is $\frac{d y}{d x}$ $dy/dx$ ?

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