Continuity and Differentiability — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Differentiate the sin–1$\left(\frac{2^{x+1}}{1+4^{x}}\right)$ w.r.t. x.
✓
Answer
Let f(x) = sin–1$\left(\frac{2^{x+1}}{1+4^{x}}\right)$. To find the domain of this function we need to find all x such that $-1 \leq \frac{2^{x+1}}{1+4^{x}} \leq 1$. Since the quantity in the middle is always positive, We need to find all x such that $\frac{2^{x+1}}{1+4^{x}} \leq$ 1, i.e., all x such that 2x + 1 $\leq$ 1 + 4x . We may rewrite this as 2 $\leq \frac{1}{2^{x}}$ + $2^x$ which is true for all x. Hence the function is defined at every real number. By putting 2x = tan $\theta$, this function may be rewritten as $f(x)=\sin ^{-1}\left[\frac{2^{x+1}}{1+4^{x}}\right]$ = $\sin ^{-1}\left[\frac{2^{x} \cdot 2}{1+\left(2^{x}\right)^{2}}\right]$ = $\sin ^{-1}\left[\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right]$ = sin–1 [sin 2$\theta$] = 2$\theta$ = 2 tan–1 (2x) Thus $f^{\prime}(x)=2 \cdot \frac{1}{1+\left(2^{x}\right)^{2}} \cdot \frac{d}{d x}\left(2^{x}\right)$ = $\frac{2}{1+4^{x}} \cdot\left(2^{x}\right) \log 2$ = $\frac{2^{x+1} \log 2}{1+4^{x}}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.