Derivative of $$$\sin^{3}{\left(x \right)}$$$

The calculator will find the derivative of $$$\sin^{3}{\left(x \right)}$$$, with steps shown.

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Find $$$\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right)$$$.


The function $$$\sin^{3}{\left(x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = u^{3}$$$ and $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 3$$$:

$${\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$

Return to the old variable:

$$3 {\color{red}\left(u\right)}^{2} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = 3 {\color{red}\left(\sin{\left(x \right)}\right)}^{2} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$

The derivative of the sine is $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

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

Thus, $$$\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right) = 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$$.


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