Chapter 4

$$ \text { Given } f(x)=|x| \text { , find } f^{\prime}(x) \text { . } $$

$$ \text { Given } f(x)=\operatorname{sgn} x \text { , find } f^{\prime}(x) $$

$$ \text { Find the derivative of } \ln |x| \text { . } $$

$$ \begin{aligned} &\text { Check the differentiability of the function \& find } f^{\prime}(x) \text { . }\\\ &f(x)=x^{2}, \quad x \geq 0\\\ &=x, \quad x<0 \end{aligned} $$

$$ \begin{aligned} &\text { Check the differentiability of the function \& find } f^{\prime}(x) \text { . }\\\ &\begin{aligned} f(x) &=\sin x, & x \geq 0 \\ &=e^{x}-1, & x<0 \end{aligned} \end{aligned} $$

$$ \begin{aligned} &\text { Given }\\\ &\begin{aligned} f(x) &=x^{2} \sin \frac{1}{x}, & & x \neq 0 \\ &=0, & & x=0 \end{aligned} \end{aligned} $$

$$ \begin{aligned} &\text { Given }\\\ &f(x)=\frac{\sin x^{2}}{x}, \quad x \neq 0\\\ &=0, \quad x=0,\\\ &\text { find } f^{\prime}(x) \text { . } \end{aligned} $$

$$ \text { Given } f(x)=\sqrt[3]{x-\sin x} \text { , find } f^{\prime}(x) $$

$$ \text { Given } f(x)=\sqrt{e^{x^{4}}}-1 \text { , find } f^{\prime}(x) $$

$$ \text { Given } f(x)=\left(x^{2}-1\right) \sin ^{-1} x, \text { find } f^{\prime}(x) $$

$$ \text { Given } f(x)=\ln x \cdot \sin ^{-1} x, \text { find } f^{\prime}(x) $$

$$ \text { Given } f(x)=\sin x \cdot \sqrt{x}, \text { find } f^{\prime}(x) $$

$$ \text { If } f(x)=\sqrt{x^{2}-2 x+1} \text { , then find } f^{\prime}(x) \text { . } $$

$$ \text { Given } f(x)=|x|+|x-1|, \text { find } f^{\prime}(x) $$

Given \(\begin{aligned} f(x) &=(x-1)^{2} \sin \left(\frac{1}{x-1}\right)-|x|, x \neq 1 \\ &=-1, \quad x=1 \end{aligned}\) Find the set of points where \(f(x)\) is not differentiable.

$$ \begin{aligned} &\text { Discuss the continuity and differentiability of the function }\\\ &\begin{aligned} f(x) &=\frac{x}{1+|x|},|x| \geq 1 \\ &=\frac{x}{1-|x|},|x|<1 \end{aligned} \end{aligned} $$

Let \(\begin{aligned} f(x) &=\sqrt{x}\left(1+x \sin \frac{1}{x}\right), & x>0 \\\ &=-\sqrt{-x}\left(1+x \sin \frac{1}{x}\right), & x<0 \\ &=0 & & x=0 \end{aligned}\) Show that \(f^{\prime}(x)\) exists everywhere and is finite except at \(x=0\).

Draw the graph of the function \(y=|x-1|+|x-2|\) in the interval \([0,3]\) and discuss the continuity and differentiability of the function in this interval. \\{Ans. continuous, not differentiable at \(x=1 \& 2\\}\)

Let \(f(x)\) be defined in the interval \([-2,2]\) such that \(f(x)=-1, \quad-2 \leq x \leq 0\) \(=x-1, \quad 0

Find \(a \& b\) such that the function \(f(x)=a x^{2}-b, \quad|x|<1\) \(=-\frac{1}{|x|}, \quad|x| \geq 1\) is continuous and differentiable function.

Test for continuity and differentiability of the function \(\&\) find \(f^{\prime}(x)\) \(f(x)=x, \quad x\) is rational \(=-x, \quad x\) is irrational.

Check continuity and differentiability of the function \(\&\) find \(f^{\prime}(x)\) \(f(x)=x^{2}, \quad x\) is rational \(=x^{3}, x\) is irrational

$$ \text { If } f(x)=e^{x} g(x), g(0)=2, g^{\prime}(0)=1, \text { then find } f^{\prime}(0) $$

$$ \text { If } f(x)=e^{x} g(x), g(0)=2, g^{\prime}(0)=1, \text { then find } f^{\prime}(0) \text { . } $$

If the derivative of the function \(\begin{aligned} f(x) &=a x^{2}+b, \quad x<-1 \\ &=b x^{2}+a x+4, \quad x \geq-1 \end{aligned}\) is everywhere continuous, then find \(a\) and \(b\).

Let \(R\) be the set of real numbers and \(f: R \rightarrow R\) such that for all \(x\) and \(y\) in \(R,|f(x)-f(y)| \leq|x-y|^{3}\). Prove that \(f(x)\) is a constant.

$$ \begin{aligned} &\text { Suppose } p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots \ldots \ldots \ldots . .+a_{n} x^{n} . \text { If }|p(x)| \leq\left|e^{x-1}-1\right| \forall x \geq 0, \text { then prove that }\\\ &\left|a_{1}+2 a_{2}+\ldots \ldots \ldots \ldots+n a_{n}\right| \leq 1 \end{aligned} $$

$$ \text { If } y=x^{2}-3 x+2, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\left(x^{2}+1\right)^{3}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\frac{a}{x^{n}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=x e^{x^{2}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\frac{1}{1+x^{3}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\sqrt{a^{2}-x^{2}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\ln \left(x+\sqrt{1+x^{2}}\right), \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\frac{1}{a+\sqrt{x}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=e^{\sqrt{x}}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\sqrt{1-x^{2}} \sin ^{-1} x, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { 1. If } y=\sin ^{-1}(a \sin x) \text { , find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=x^{x}, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=e^{2 x-1}, \text { find }\left(\frac{d^{2} y}{d x^{2}}\right)_{x=0} $$

$$ \text { If } y=\tan ^{-1} x, \text { find }\left(\frac{d^{2} y}{d x^{2}}\right)_{x=1} $$

$$ \text { If } y=\cos ^{2} x, \text { find } \frac{d^{3} y}{d x^{3}} $$

$$ \text { If } y=(x+10)^{6}, \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{x=2} $$

$$ \text { If } f(x)=\ln (\ln x), \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{x=e} $$

$$ \text { If } y=x^{3} \ln x, \text { find } \frac{d^{4} y}{d x^{4}} $$

$$ \text { If } y=a \sin 2 x, \text { find } \frac{d^{4} y}{d x^{4}} \text { . } $$

$$ \text { If } y=\frac{1}{1-x}, \text { find } \frac{d^{5} y}{d x^{5}} $$

$$ \text { If } y=\frac{1-x}{1+x}, \text { find }\left(\frac{d^{5} y}{d x^{5}}\right)_{x=1} $$

$$ \text { If } a x^{2}+2 h x y+b y^{2}=1, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}, \text { find } \frac{d^{2} y}{d x^{2}} $$