# Exercise ID258

Mathematical analysis → Derivation → Derivatives of composite functions
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Compute the derivative of the given function below.

$y=\mathrm{ln}\left(\mathrm{cos}x·{\mathrm{sin}}^{2}2x\right)$

$y\text{'}=\frac{1}{g}·g\text{'}\phantom{\rule{0ex}{0ex}}$ $g\text{'}=u\text{'}·v+u·v\text{'}\phantom{\rule{0ex}{0ex}}$ $g\text{'}={-}{s}{i}{n}{}{x}{·}{s}{i}{{n}}^{2}{2}{x}+{c}{o}{s}{}{x}·{4}{·}{s}{i}{n}{}{2}{x}{·}{c}{o}{s}{2}{x}\phantom{\rule{0ex}{0ex}}$ $y\text{'}=\frac{1}{\mathrm{cos}x{\mathrm{sin}}^{2}2x}·\left(-\mathrm{sin}x·{\mathrm{sin}}^{2}2x+\mathrm{cos}x·4·\mathrm{sin}2x·\mathrm{cos}2x\right)\phantom{\rule{0ex}{0ex}}$

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]=\frac{df}{dg}·\frac{dg}{dx}=f{\text{'}}_{g}·g{\text{'}}_{x}$

$\frac{d}{dx}\left(\mathrm{ln}x\right)=\frac{1}{x}$

$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$

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

$\frac{d}{dx}\left(\mathrm{cos}x\right)=-\mathrm{sin}x$