Question

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

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

Answer 1

Answer: (D)

$10$

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

Using chain rule of differentiation

$\frac{dy}{dx}=\frac{d}{d(x^2+2)}\left[f\right(x^2+2\left)\right]\cdot \frac{d}{dx}(x^2+2)$

$=f^{\prime }(x^2+2)\cdot 2x=2x{f}^{\prime }(x^2+2)$

Hence $\frac{dy}{dx}{|}_{x=1}=2\cdot 1\cdot f^{\prime }(1^2+2)=2\cdot f^{\prime }\left(3\right)=2\cdot 5=10$