Derivative
(2) Find the derivatives of the following w. r. t. \(x\). at the points indicated against them by using method of first principle
(a) \(\sqrt{2 x+5}\) at \(x=2\)
(b) \(\tan x\) at \(x=\pi / 4\)
(c) \(2^{3 x+1}\) at \(x=2\)
(d) \(\log (2 x+1)\) at \(x=2\)
(e) \(e^{3 x-4}\) at \(x=2\)
(f) \(\cos x\) at \(x=\frac{5 \pi}{4}\)
(3) Show that the function \(\mathrm{f}\) is not differentiable at \(x=-3\), where \(f(x)=x^{2}+2\) for \(x<-3\) \(=2-3 x\) for \(x \geq-3\)
(4) Show that \(f(x)=x^{2}\) is continuous and differentiable at \(x=0\).
(5) Discuss the continuity and differentiability of
(i) \(f(x)=x|x|\) at \(x=0\)
(ii) \(f(x)=(2 x+3)|2 x+3|\) at \(x=-3 / 2\)
(6) Discuss the continuity and differentiability of \(f(x)\) at \(x=2\) \(f(x)=[x]\) if \(x \in[0,4)\). \(\quad[\) where \([*]\) is a greatest integer (floor) function]
(7) Test the continuity and differentiability of \(f(x)=3 x+2\) if \(x>2\) \(=12-x^{2}\) if \(x \leq 2\) at \(x=2 .\)
(8) If \(f(x)=\sin x-\cos x\) if \(x \leq \pi / 2\) \(=2 x-\pi+1\) if \(x>\pi / 2\). Test the continuity and differentiability of \(\mathrm{f}\) at \(x=\pi / 2\)
(9) Examine the function \(f(x)=x^{2} \cos \left(\frac{1}{x}\right)\), for \(x \neq 0\) \(=0, \quad\) for \(x=0\) for continuity and differentiability at \(x=0\).
(a) \(\sqrt{2 x+5}\) at \(x=2\)
(b) \(\tan x\) at \(x=\pi / 4\)
(c) \(2^{3 x+1}\) at \(x=2\)
(d) \(\log (2 x+1)\) at \(x=2\)
(e) \(e^{3 x-4}\) at \(x=2\)
(f) \(\cos x\) at \(x=\frac{5 \pi}{4}\)
(3) Show that the function \(\mathrm{f}\) is not differentiable at \(x=-3\), where \(f(x)=x^{2}+2\) for \(x<-3\) \(=2-3 x\) for \(x \geq-3\)
(4) Show that \(f(x)=x^{2}\) is continuous and differentiable at \(x=0\).
(5) Discuss the continuity and differentiability of
(i) \(f(x)=x|x|\) at \(x=0\)
(ii) \(f(x)=(2 x+3)|2 x+3|\) at \(x=-3 / 2\)
(6) Discuss the continuity and differentiability of \(f(x)\) at \(x=2\) \(f(x)=[x]\) if \(x \in[0,4)\). \(\quad[\) where \([*]\) is a greatest integer (floor) function]
(7) Test the continuity and differentiability of \(f(x)=3 x+2\) if \(x>2\) \(=12-x^{2}\) if \(x \leq 2\) at \(x=2 .\)
(8) If \(f(x)=\sin x-\cos x\) if \(x \leq \pi / 2\) \(=2 x-\pi+1\) if \(x>\pi / 2\). Test the continuity and differentiability of \(\mathrm{f}\) at \(x=\pi / 2\)
(9) Examine the function \(f(x)=x^{2} \cos \left(\frac{1}{x}\right)\), for \(x \neq 0\) \(=0, \quad\) for \(x=0\) for continuity and differentiability at \(x=0\).
