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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Differentiate the following w.r.t. x:
$(\sin\text{x})^{\cos\text{x}}$
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Answer

Let $\text{y}=(\sin\text{x})^{\cos\text{x}}$
$\Rightarrow\ \log\text{y}=\log(\sin\text{x})^{\cos\text{x}}=\cos\text{x}\log(\sin\text{x})$
Differentiate both sides w.r.t.x, we get
$\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{d}}{\text{dx}}=\cos\text{x}\cdot\frac{\text{d}}{\text{dx}}(\log\sin\text{x})+\log\sin\text{x }\frac{\text{d}}{\text{dx}}(\cos\text{x})$
$=\cos\text{x}\cdot\frac{1}{\sin\text{x}}\cdot\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}\cdot(-\sin\text{x})$
$=\cos\text{x}\cdot\cos\text{x}-\log(\sin\text{x})\cdot\sin\text{x}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\text{y}\big[\cot\text{x}\cos\text{x}-\sin\text{x}\cdot\log(\sin\text{x})\big]$
$=(\sin\text{x})^{\cos\text{x}}\big[\cot\text{x}\cos\text{x}-\sin\text{x}\cdot\log(\sin\text{x})\big]$

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