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  • Differentiate each of the following w.r.t. x 2 ^ cos^2 x

Differentiate each of the following w.r.t. x
2 ^{\cos ^2 x}

Answers (1)

Given: 2 ^{\cos ^2 x}

Let Assume y=2 ^{\cos ^2 x}

Now, Taking Log on both sides we get,

\begin{aligned} &\log y=\log 2^{\cos ^{2} x}\\ &\log \mathrm{y}=\cos ^{2} \mathrm{x} \cdot \log 2\\ &\text { Now, Differentiate w.r.t } x\\ &\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}\left[\cos ^{2} x \cdot \log 2\right]\\ &\frac{1}{y} \frac{d y}{d x}=[2 \cos x \cdot(-\sin x) \cdot \log 2]\\ &\frac{d y}{d x}=y[2 \cos x \cdot(-\sin x) \cdot \log 2] \end{aligned}

Now, substitute the value of y

\\ \frac{\mathrm{dy}}{\mathrm{dx}}=-2^{\cos ^{2} x}[2 \cos \mathrm{x} \cdot(\sin \mathrm{x}) \cdot \log 2] \\ \text { Hence, } \frac{\mathrm{dy}}{\mathrm{dx}}=-2^{\cos ^{2} x}[2 \cos \mathrm{x} \cdot(\sin \mathrm{x}) \cdot \log 2]

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