Question

If $y=f\left(x^2+2\right)$ and $f^{\text{'}}\left(3\right)=5$, then $\frac{dy}{dx}$at $x=1$ is

• A. $5$
• B. $25$
• C. $15$
• D. $10$

Answer 1

The correct option is D

$10$

Finding the derivative:

Given that, $y=f\left(x^2+2\right)$ and $f^{\text{'}}\left(3\right)=5$.

Differentiate $y$ with respect to $x$using chain rule.

$\Rightarrow \frac{dy}{dx}=f\text{'}\left(x^2+2\right)·\left(2x\right)$

At $x=1$

$\begin{array}{rcl}\frac{dy}{dx}& =& f\text{'}\left(1+2\right)·2\\ & =& 2f\text{'}\left(3\right)\\ & =& 2·5\\ & =& 10\end{array}$

Hence, option D is correct.