How do you differentiate ${sin}^{3} (x) cos (x)$ $sin^3(x)cos(x)$ ?

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Answer 1 · 2016-05-20T11:17:29

$3 {sin}^{2} x . {cos}^{2} x - {sin}^{4} x$ $3sin^2x . cos^2x-sin^4x$

$\frac{d}{d x} ({sin}^{3} (x) cos (x))$ $\frac{d}{dx}(\sin ^3(x)\cos (x))$

$(f\cdot g)^'=f^'\cdot g+f\cdot g^'$

$f=\sin ^3(x),\g=\cos(x)$

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

We have,
$\frac{d}{dx}(\sin ^3(x))=3\sin ^2(x)\cos(x)$
Also,
$\frac{d}{dx}(\cos (x))=-\sin (x)$

$=3\sin ^2(x)\cos(x)\cos (x)+(-\sin (x))\sin ^3(x)$

Simplifying it,we get,
$=3\sin ^2(x)\cos ^2(x)-\sin ^4(x)$

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