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  4. If y = f (x2+2) and f '(3)=5, then (dy/dx) at x=1 is

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

KEAMKEAM 2014Limits and Derivatives

Solution:

Given, $y=f\left(x^{2}+2\right)$
On differentiating both sides w.r.t. $x$, we get
$\frac{d y}{d x}=f^{'}\left(x^{2}+2\right) \times 2 x$
Put $x=1$, we get
$\frac{d y}{d x} =f^{'}\left(1^{2}+2\right) \times 2=f^{\prime}(3) \times 2 $
$=5 \times 2 \,\,\,\left[\because f^{'}(3)=5, \text { given }\right] $
$ =10 $