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  4. Let f(x + y) = f(x) f(y) and f(x) = 1 + sin (3x) g(x), where g is differentiable. The f ′(x) is equal to

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Q. Let $f(x + y) = f(x) f(y) $ and $f(x) = 1 + \sin \, (3x) g(x)$, where $g$ is differentiable. The $f ′(x)$ is equal to

KEAMKEAM 2017Continuity and Differentiability

Solution:

$f^{'}(x) =\displaystyle \lim _{h \rightarrow 0} \frac{f(x +h)-f(x)}{h} $
$=\displaystyle \lim _{h \rightarrow 0} \frac{f(x) f(h)-f(x)}{h} $
$=f(x) \displaystyle \lim _{h \rightarrow 0}\left(\frac{1+\sin 3 h(g(h))-1)}{h}\right) $
$=f(x) \displaystyle \lim _{h \rightarrow 0} \frac{\sin 3 h}{3 h} \displaystyle \lim _{h \rightarrow 0}\, g(h) $
$=f(x) \times 1 \times g(0)=f(x) g(0) $