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: $(\log\text{x})^{\cos\text{x}}$
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Answer
Let $\text{y}=(\log\text{x})^{\cos\text{x}}\ .....(\text{i})$ Taking log on both the sides, $\log\text{y}=(\log\text{x})^{\cos\text{x}}$ $\Rightarrow\log\text{y}=\cos\text{x}\log(\log\text{x})$ Differentiating with respect to x, $\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\cos\text{x}\frac{\text{d}}{\text{dx}}\log(\log\text{x})+\log(\log\text{x})\frac{\text{d}}{\text{dx}}(\cos\text{x})$ $\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\cos\text{x}}{\log\text{x}}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log(\log\text{x})\times(-\sin\text{x})$ $\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\cos\text{x}}{\log\text{x}}\times\big(\frac{1}{\text{x}}\big)-\sin\text{x}\log(\log\text{x})$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{\cos\text{x}}{\text{x}\log\text{x}}-\sin\text{x}\log(\log\text{x})\Big]$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=(\log\text{x}^{\cos\text{x}})\Big[\frac{\cos\text{x}}{\text{x}\log\text{x}}-\sin\text{x}\log(\log\text{x})\Big]$ [Using equation (i)]
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