Limit Continuity and Differentiability 7 Question 18
18. Which of the following functions is differentiable at $x=0$ ?
(2001, 2M)
(a) $\cos (|x|)+|x|$
(b) $\cos (|x|)-|x|$
(c) $\sin (|x|)+|x|$
(d) $\sin (|x|)-|x|$
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Answer:
Correct Answer: 18. (b)
Solution:
- Given, $x^{2}+y^{2}=1$
On differentiating w.r.t. $x$, we get
$$ \begin{array}{rlrl} & & 2 x+2 y y^{\prime} & =0 \\ \Rightarrow & x+y y^{\prime} & =0 . \end{array} $$
Again, differentiating w.r.t. $x$, we get
$$ \begin{aligned} & 1+y^{\prime} y^{\prime}+y y^{\prime \prime}=0 \\ & \Rightarrow \quad 1+\left(y^{\prime}\right)^{2}+y y^{\prime \prime}=0 \end{aligned} $$
