CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Differentiate the following functions with respect to x: $(\sin\text{x})^{\log\text{x}}$
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Answer
Let $\text{y}=(\sin\text{x})^{\log\text{x}}\ .....(\text{i})$ Taking log on both sides, $\log\text{y}=\log(\sin\text{x})^{\log\text{x}}$ $\Rightarrow\log\text{y}=\log\text{x}\log\sin\text{x}$ Differentiating with respect to x, $\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log\text{x}\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})$ $\Rightarrow \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\log\text{x}\frac{1}{\sin\text{x}}\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}\Big(\frac{1}{\text{x}}\Big)$ $\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\log\text{x}}{\sin\text{x}}(\cos\text{x})+\frac{\log\sin\text{x}}{\text{x}}$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\log\text{x}\cot\text{x}+\frac{\log\sin\text{x}}{\text{x}}\Big]$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=(\sin\text{x})^{\log\text{x}}\Big[\log\text{x}\cot\text{x}+\frac{\log\sin\text{x}}{\text{x}}\Big]$ [Using equation (i)]
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