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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sin\text{x}}{1+\cos\text{x}}\Big),\pi<\text{x}<\pi$
✓

Answer

Let $\text{f(x)}=\tan^{-1}\Big(\frac{\sin\text{x}}{1+\cos\text{x}}\Big)$
This function is defined for all real numbers where $\cos\text{x}\neq1$
$\text{f(x)}=\tan^{-1}\Big(\frac{\sin\text{x}}{1+\cos\text{x}}\Big)$
$\Rightarrow\ \text{f(x)}=\tan^{-1}\Bigg[\frac{2\sin\big(\frac{\text{x}}{2}\big)\cos\big(\frac{\text{x}}{2}\big)}{2\cos^2\big(\frac{\text{x}}{2}\big)}\Bigg]$
$\Rightarrow\ \text{f(x)}=\tan^{-1}\big[\tan\big(\frac{\text{x}}{2}\big)\big]=\frac{\text{x}}{2}$
Thus, $\text{f'(x)}=\frac{\text{d}}{\text{dx}}\big(\frac{\text{x}}{2}\big)=\frac{1}{2}$

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