Question

Let $f^{\prime }\left(x\right)=\sin x^2$ and $y=f(x^2+1)$ then $\frac{dy}{dx}$ at $x=2$ is

• A. $0$
• B. $1$
• C. $4\sin 5$
• D. $4\sin 25$

Answer 1

The correct option is D $4\sin 25$

$y=f(x^2+1)$

$\frac{dy}{dx}=f^{\prime }(x^2+1)\times 2x$

$\frac{dy}{dx}$ at $x=2$

$=f^{\prime }(2^2+1)\times 2\left(2\right)=4f^{\prime }\left(5\right)$

$f^{\prime }\left(x\right)=\sin x^2$

$\therefore f^{\prime }\left(5\right)=\sin 5^2=\sin 25$

$\therefore \frac{dy}{dx}$ at $x=2$

$=4f^{\prime }\left(5\right)=4\sin 25$