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# What is the derivative of sinx - (1/3)sinx^3?

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#### Explanation

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#### Explanation:

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Jim G. Share
Jun 23, 2018

$\cos x - {x}^{2} \cos {x}^{3}$

#### Explanation:

$\text{differentiate "sinx^3" using the "color(blue)"chain rule}$

$\text{given "y=f(g(x))" then}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(g \left(x\right)\right) \times g ' \left(x\right) \leftarrow \textcolor{b l u e}{\text{chain rule}}$

$\frac{d}{\mathrm{dx}} \left(\sin {x}^{3}\right) = \cos {x}^{3} \times \frac{d}{\mathrm{dx}} \left({x}^{3}\right) = 3 {x}^{2} \cos {x}^{3}$

$\Rightarrow \frac{d}{\mathrm{dx}} \left(\sin x - \frac{1}{3} \sin {x}^{3}\right)$

$= \cos x - \frac{1}{3} \times 3 {x}^{2} \cos {x}^{3}$

$= \cos x - {x}^{2} \cos {x}^{3}$

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