Limit Continuity and Differentiability 7 Question 34
35. Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in R$. If $f(x)$ is differentiable at $x=0$, then
(2011)
(a) $f(x)$ is differentiable only in a finite interval containing zero
(b) $f(x)$ is continuous for all $x \in R$
(c) $f^{\prime}(x)$ is constant for all $x \in R$
(d) $f(x)$ is differentiable except at finitely many points
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Answer:
Correct Answer: 35. $(a=1)$
Solution:
- Since, $y=e^{x \sin x^{3}}+(\tan x)^{x}$, then
$y=u+v$, where $u=e^{x \sin x^{3}}$ and $v=(\tan x)^{x}$
$$ \Rightarrow \quad \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x} $$
Here, $u=e^{x \sin x^{3}}$ and $\log v=x \log (\tan x)$
On differentiating both sides w.r.t. $x$, we get
$$ \begin{aligned} \frac{d u}{d x} & =e^{x \sin x^{3}} \cdot\left(3 x^{3} \cos x^{3}+\sin x^{3}\right) \\ \text { and } \quad \frac{1}{v} \cdot \frac{d v}{d x} & =\frac{x \cdot \sec ^{2} x}{\tan x}+\log (\tan x) \\ \frac{d v}{d x} & =(\tan x)^{x}[2 x \cdot \operatorname{cosec}(2 x)+\log (\tan x)] \ldots \text { (iii) } \end{aligned} $$
From Eqs. (i), (ii) and (iii), wet get
$$ \frac{d y}{d x}=e^{x \sin x^{3}}\left(3 x^{3} \cdot \cos x^{3}+\sin x^{3}\right)+(\tan x)^{x} $$
$[2 x \operatorname{cosec} 2 x+\log (\tan x)]$
