Question

If $f^{\prime }\left(x\right)=\sin \left(\log x\right)$ and $y=f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ is :

• A. $\frac{9\cos \left(\log x\right)}{x(3-2x{)}^2}$
• B. $\frac{9\cos \left(\frac{2x+3}{3-2x}\right)}{x(3-2x{)}^2}$
• C. $\frac{9\sin \left(\frac{2x+3}{3-2x}\right)}{x(3-2x{)}^2}$
• D. None of the above

Answer 1

The correct option is D None of the above

$f^{\prime }\left(x\right)=\sin \left(\log x\right)$ and $y=f\left(\frac{2x+3}{3-2x}\right)$

Let $y=f\left(t\right),$ where $t=\frac{2x+3}{3-2x}$
$\Rightarrow \frac{dy}{dx}=f^{\prime }\left(t\right)\times \frac{dt}{dx}$

$=\sin \left(\log t\right)\times \frac{(3-2x)2-(2x+3)(-2)}{(3-2x{)}^2}$

$=\sin \left(\log \frac{2x+3}{3-2x}\right)\times \frac{12}{(3-2x{)}^2}$