Let f(x) be a real valued function, then its 
Left Hand Derivative (L.H.D.) : \(\begin{equation} \mathrm{L} f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h} \end{equation}\) 
Right Hand Derivative (R.H.D.) : \(\begin{equation} \mathrm{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \end{equation}\) 
Also, a function jfx) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal 
For the function \(\begin{equation} f(x)=\left\{\begin{array}{l} |x-3|, x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, x<1 \end{array}\right. \end{equation}\) answer the following questions
(i) R.H.D. of f(x) at x = 1is

(ii) L.H.D. of f(x) at x = 1 is

(iii) f(x) is non-differentiable at

(iv) Find the value of f'(2).

(v) The value of f'( -1) is

Let x = f(t) and y = get) be parametric forms with t as a parameter,
then \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \end{equation}\) ,where \(\begin{equation} f^{\prime}(t) \neq 0 \end{equation}\) \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \end{equation}\) where \(\begin{equation} f^{\prime}(t) \neq 0 \end{equation}\).
(i) The derivative off (tanx) w.r.t. \(\begin{equation} g(\sec x) \text { at } x=\frac{\pi}{4} \end{equation}\) ,where f'(1) and \(\begin{equation} g^{\prime}(\sqrt{2})=4 \end{equation}\) is

(ii) The derivate of \(\begin{equation} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) \end{equation}\) ,with respect to \(\begin{equation} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) \end{equation}\) is

(iii) The derivative of \(\begin{equation} e^{x^{3}} \end{equation}\) with respect to log x is

(iv) The derivative of \(\begin{equation} \cos ^{-1}\left(2 x^{2}-1\right) \end{equation}\) w.r.t. cos^-1x is

(v) If \(\begin{equation} y=\frac{1}{4} u^{4} \end{equation}\)  and \(\begin{equation} u=\frac{2}{3} x^{3}+5 \end{equation}\) then \(\begin{equation} \frac{d y}{d x}= \end{equation}\)

Let \(\begin{equation} f: A \rightarrow B \end{equation}\) and \(\begin{equation} g: B \rightarrow C \end{equation}\) be two functions defined on non-empty sets A, B, C,
then \(\begin{equation} \text { gof }: A \rightarrow C \end{equation}\)  be is called the composition off and g defined as, \(\begin{equation} g o f(x)=g\{f(x)\} \forall x \in A \end{equation}\) .
Consider the functions \(\begin{equation} f(x)=\left\{\begin{array}{ll} \sin x, & x \geq 0 \\ 1-\cos x, & x \leq 0 \end{array}, g(x)=e^{x}\right. \end{equation}\) and
then answer the following questions. 
(i) The function gof(x) is defined as

(ii) \(\begin{equation} \frac{d}{d x}\{\operatorname{gof}(x)\}= \end{equation}\) 

(iii) R.H.D. of gof(x) at x = 0 is

(iv) L.H.D. of gof(x) at x = 0 is

(v) The value of \(\begin{equation} f^{\prime}(x) \text { at } x=\frac{\pi}{4} \end{equation}\) is 

The function f(x) will be discontinuous at x = a if f(x) has 
(a) Discontinuity of first kind \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist but are not equal. If is also known as irremovable discontinuity.
(b)  Discontinuity of second kind: If none of the limits \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) exist.
(c) Removable discontinuity: \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist and equal but not equal to f(a). 
Based on the above information, answer the following questions.
(i) If \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3}, & \text { for } x \neq 3 \\ 4, & \text { for } x=3 \end{array}\right. \end{equation}\) ,then at x= 3

(ii) Let \(\begin{equation} f(x)=\left\{\begin{array}{ll} x+2, & \text { if } x \leq 4 \\ x+4, & \text { if } x>4 \end{array}\right. \end{equation}\) ,then at x = 4

(iii) Consider the function f(x) defined \(\begin{equation} f(x)=\left\{\begin{array}{l} \frac{x^{2}-4}{x-2} \\ 5 \end{array}\right. \end{equation}\), for \(\begin{equation} x \neq 2 \end{equation}\) 

 (iv) If \(\begin{equation} f(x)=\left\{\begin{array}{cc} \frac{x-|x|}{x}, & if\ x \neq 0 \\ 2, & if\ x=0 \end{array}\right. \end{equation}\) ,then x = 0

(v) If \(\begin{equation} f^{\prime}(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log (1+2 x)}, & \text { if } x \neq 0 \\ 7, & \text { if } x=0 \end{array}\right. \end{equation}\), then at x = 0

If a real valued function f(x) is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
For example, every polynomial. constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
Based on the above information, answer the following questions.
(i) If \(\begin{equation} f(x)=\left\{\begin{array}{l} x, \text { for } x \leq 0 \\ 0, \text { for } x>0 \end{array}\right. \end{equation}\) , then at x = 0

(ii) If \(\begin{equation} f(x)=|x-1|, x \in R \end{equation}\) ,then at x= 1 

(iii) f(x) = x³ is

(iv) f(x) = [sin x], then which of the following is true? 

(v) If f(x) = sin^-1x, \(\begin{equation} -1 \leq x \leq 1 \end{equation}\), then