Differentiate the following w.r.t. x: ( x+1 ) - Vidyadip

Question

Differentiate the following w.r.t. x:
$\cos\big(\tan\sqrt{\text{x}+1}\big)$

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Answer

Let $\text{y}=\cos\big(\tan\sqrt{\text{x}+1}\big)$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\cos\big(\tan\sqrt{\text{x}+1}\big)$ $=-\sin\big(\tan\sqrt{\text{x}+1}\big)\cdot\frac{\text{d}}{\text{dx}}\big(\tan\sqrt{\text{x}+1}\big)$
$=-\sin\big(\tan\sqrt{\text{x}+1}\big).\sec^2\sqrt{\text{x}+1}.\frac{\text{d}}{\text{dx}}(\text{x}+1)^{\frac{1}{2}}$ $\Big[\because\frac{\text{d}}{\text{dx}}(\tan\text{x})=\sec^2\text{x}\Big]$
$=-\sin\big(\tan\sqrt{\text{x}+1}\big).\big(\sec\sqrt{\text{x}+1}\big)^2.\frac{1}{2}(\text{x}+1)^{\frac{1}{2}}.\frac{\text{d}}{\text{dx}}(\text{x}+1)$
$=\frac{-1}{2\sqrt{\text{x}+1}}.\sin\big(\tan\sqrt{\text{x}+1}\big).\sec^2\big(\sqrt{\text{x}+1}\big)$

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