Question

If $f \left(x\right)=\frac{5x}{\left(1-x\right)^{2/3}}+cos^{2}\left(2x+1\right)$, then $f'(0) =$

• A. $5 + 2 \,sin \,2$
• B. $5 + 2 \,cos \,2$
• C. $5 - 2\,sin \,2$
• D. $5 - 2 \,cos \,2$

Answer 1

Answer: (C)

$f \left(x\right)=5x\left(1-x\right)^{-\frac{2}{3}}+cos^{2}\left(2x+1\right)$
$f '\left(x\right)=5\left\{x\times\frac{-2}{3}\left(1-x\right)^{-5/3}\left(-1\right)+\left(1-x\right)^{-2/3}\times1\right\}$
$+2\,cos\left(2x+1\right)\left\{-sin\left(2x+1\right)\times2\right\}$
$\Rightarrow f '\left(x\right)=5\left(1-x\right)^{-\frac{2}{3}}+\frac{10x}{3}\left(1-x\right)^{-\frac{5}{3}}-2\,sin\left(4x+2\right)$
$\therefore f '\left(0\right)=5-2\,sin\,2$.