Calculating derivatives Find the derivative of the following functions. y = cos^2 x

Calculating derivatives Find the derivative of the following functions.
$y={\mathrm{cos}}^{2}x$
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Step 1
We have to find derivatives:
$y={\mathrm{cos}}^{2}x$
We know the derivatives formula,
$\frac{{dx}^{n}}{dx}=n{x}^{n-1}$
$\frac{d{\left(f\left(x\right)\right)}^{2}}{dx}=2{\left(f\left(x\right)\right)}^{2-1}\frac{df\left(x\right)}{dx}$
$=2f\left(x\right){f}^{\prime }\left(x\right)$
$\frac{d\mathrm{cos}x}{dx}=-\mathrm{sin}x$
Here
$f\left(x\right)=\mathrm{cos}x$
Step 2
Differentiating given function with respect to 'x', we get
$y={\mathrm{cos}}^{2}x$
$\frac{dy}{dx}=\frac{d{\mathrm{cos}}^{2}x}{dx}$
$=2{\left(\mathrm{cos}x\right)}^{2-1}\frac{d\mathrm{cos}x}{dx}$
$=2\mathrm{cos}x\left(-\mathrm{sin}x\right)$
$=-2\mathrm{sin}x\mathrm{cos}x$
$=-2\mathrm{sin}2x$
Hence, derivatives of given function is $-\mathrm{sin}2x$.