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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:
$2^{\cos^2}\text{x}$
✓

Answer

Let $\text{y}=2^{\cos^2}\text{x}$
Taking logarithm on both sides, we get
$\log\text{y}=\log2\cos^2\text{x}$
$\Rightarrow\ \frac{\text{d}}{\text{dy}}(\log\text{y})\cdot\frac{\text{dy}}{\text{dx}}=\log2\frac{\text{d}}{\text{dx}}(\cos^2\text{x})$
$\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\log2(2\cos\text{x})\frac{\text{d}}{\text{dx}}\cos\text{x}$
$\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\log2\cdot2\cos\text{x}\cdot(-\sin\text{x})$
$\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\log2\cdot[-(\sin2\text{x})]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=-\text{y}\cdot\log2(\sin2\text{x})$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=-2^{\cos^2}\text{x}\cdot\log2(\sin2\text{x})$ $\big[\because\text{y}=2^{\cos^2}\text{x}\big]$

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