Limit Continuity And Differentiability Ques 14
- If $x e^{x y}=y+\sin ^2 x$, then at $x=0, \frac{d y}{d x}= ………..$ $\qquad$
(1996, 2M)
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Answer:
Correct Answer: 14.$(1)$
Solution: Given, $x e^{x y}=y+\sin ^2 x$ $\quad$ ……..(i)
On putting $x=0$, we get
$ \begin{aligned} 0 \cdot e^0 & =y+0 \\ y & =0 \end{aligned} $
On differentiating Eq. (i) both sides w.r.t. $x$, we get
$ 1 \cdot e^{x y}+x \cdot e^{x y}\left(x \cdot \frac{d y}{d x}+y\right)=\frac{d y}{d x}+2 \sin x \cos x $
On putting $x=0, y=0$, we get
$ \begin{aligned} e^0+0(0+0) & =\left[\frac{d y}{d x}\right] _{(0,0)}+2 \sin 0 \cos 0 \\ \Rightarrow \quad\left[\frac{d y}{d x}\right] _{0,0} & =1 \end{aligned} $
BETA
