Continuity and Differentiability — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Differentiate the $\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)$ w.r.t. x.
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Answer
Let f(x) = tan–1$\left(\frac{\sin x}{1+\cos x}\right)$. Observe that this function is defined for all real numbers, where cos x $\neq$ – 1; i.e., at all odd multiples of $\pi$. We may rewrite this function as f(x) = tan-1$\left(\frac{\sin x}{1+\cos x}\right)$ = tan-1$\left[\frac{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}{2 \cos ^{2} \frac{x}{2}}\right]$ f(x) = tan-1$\left[\tan \left(\frac{x}{2}\right)\right]=\frac{x}{2}$ $\Rightarrow$ $f^\prime(x)= \frac12$
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