MathematicsIf f(1)=3, f'(1)=2,then ddx{logf(ex+2x)} at x=0 is _____________

If $f (1) = 3, f^{'} (1) = 2,$ then $\frac{d}{d x} {\log f (e^{x} + 2 x)}$ at $x = 0$ is _____________

Solution:

Let $y = \frac{d}{d x} {\log f (e^{x} + 2 x)} = \frac{f^{'} (e^{x} + 2 x) (e^{x} + 2)}{f (e^{x} + 2 x)}$

$∴ (y)_{x = 0} = \frac{1}{f (1)} \cdot f^{'} (1) \cdot 3 = \frac{2}{3} \cdot 3 = 2 .$

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