Limit Continuity And Differentiability Ques 29
- Let $y=e^{x \sin x^3}+(\tan x)^x$, find $\frac{d y}{d x}$.
(1981, 2M)
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Answer:
Correct Answer: 29.$(e^{x \sin x^3}\left(3 x^3 \cos x^3+\sin x^3\right)+(\tan x)^x[2 x \operatorname{cosec} 2 x+\log (\tan x)])$
Solution: Since, $y=e^{x \sin x^3}+(\tan x)^x$, then
$y=u+v$, where $u=e^{x \sin x^3}$ and $v=(\tan x)^x$
$\Rightarrow \quad \frac{d y}{d x}=\left(\frac{d u}{d x}+\frac{d v}{d x}\right)$ $\quad$ ……..(i)
Here, $u=e^{x \sin x^3}$ and $\log v=x \log (\tan x)$
On differentiating both sides w.r.t. $x$, we get
$ \frac{d u}{d x}=e^{x \sin x^3} \cdot\left(3 x^3 \cos x^3+\sin x^3\right) $ $\quad$ ……..(ii)
and $\frac{1}{v} \cdot \frac{d v}{d x}=\frac{x \cdot \sec ^2 x}{\tan x}+\log (\tan x)$
$ \frac{d v}{d x}=(\tan x)^x[2 x \cdot \operatorname{cosec}(2 x)+\log (\tan x)] $ $\quad$ ……..(iii)
From Eqs. (i), (ii) and (iii), wet get
$ \begin{array}{r} \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 xz\log (\tan x)]} \end{array} $