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  • Differentiate each of the following w.r.t. x sin^mx . cos^nx

Differentiate each of the following w.r.t. x
sin^mx . cos^nx

Answers (1)

It is given sin^mx . cos^nx

y=sin^mx . cos^nx

Taking log both side, we get
\\log \mathrm{y}=\log \sin ^{\mathrm{m}} \mathrm{x} \cdot \cos ^{\mathrm{n}} \mathrm{x} \\\log \mathrm{y}=\log \sin ^{\mathrm{m}} \mathrm{x}+\log \cos ^{\mathrm{n}} \mathrm{x} \\\log y=m \log \sin x+n \log \cos x
Differentiate w.r.t x
\frac{1}{y} \frac{d y}{d x}=m \cdot \frac{1}{\sin x} \frac{d}{d x}(\sin x)+n \frac{1}{\cos x} \frac{d}{d x}(\cos x)

\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=m \cdot \frac{\cos x}{\sin x}+n \cdot \frac{-\sin x}{\cos x}\\ &\frac{1}{y} \frac{d y}{d x}=m \cdot \cot x-n \cdot \tan x\\ &\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}[\mathrm{m} \cdot \cot \mathrm{x}-\mathrm{n} \cdot \tan \mathrm{x}]\\ &\text { Substitute the value of } y \text { we get, }\\ &\frac{d y}{d x}=\sin ^{m} x \cdot \cos ^{n} x[m \cdot \cot x-n \cdot \tan x] \end{aligned}

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