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

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

(a) A function f(x) is said to be continuous in an open interval (a, b), if it is continuous at every point in this interval.
(b) A function f(x) is said to be continuous in the closed interval [a, b], if f(x) is continuous in (a, b) and \(\begin{equation} \lim _{h \rightarrow 0} f(a+h)=f(a) \text { and } \lim _{h \rightarrow 0} f(b-h)=f(b) \end{equation}\) 
If function  \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{\sin (a+1) x+\sin x}{x} & , x<0 \\ c & , x=0 \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}} & , x>0 \end{array}\right. \end{equation}\) is continuous at x = 0, then answer the following questions.
(i) The value of a is

(ii) The value of b is

(iii) The value of c is

(iv) The value of a + c is

(v) The value oi c - a is

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f[g(x)] is a differentiable function of x and \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x} \end{equation}\). This rule is also known as CHAIN RULE.
Based on the above information, find the derivative of functions w.r.t. x in the following questions
(i) \(\begin{equation} \cos \sqrt{x} \end{equation}\) 

(ii) \(\begin{equation} 7^{x+\frac{1}{x}} \end{equation}\) 

(iii) \(\begin{equation} \sqrt{\frac{1-\cos x}{1+\cos x}} \end{equation}\) 

(v) (d) :\(\begin{equation} \sec ^{-1} x+\operatorname{cosec}^{-1} \frac{x}{\sqrt{x^{2}-1}} \end{equation}\)  

If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x.
When a given relation expresses y as an implicit function of x and we want to find \(\begin{equation} \frac{d y}{d x} \end{equation}\).then
we differentiate every term of the given relation w.r.t. x. remembering that a term in y is first differentiated w.r.t. y and then multiplied by \(\begin{equation} \frac{d y}{d x} \end{equation}\).
Based on the above information, find the value of \(\begin{equation} \frac{d y}{d x} \end{equation}\) in each of the following questions 
(i) x³+x²y+xy²+y3=81

(ii) x^y = c- y

(iii) e^siny = xy

(iv) sin² x + cos²y = 1

(v) \(\begin{equation} y=(\sqrt{x})^{\sqrt{x}} \end{equation}\)