Class 12: Differentiability

$\displaystyle 1.\ \text{A real valued function }f(x)\text{ defined on }(a,b)\text{ is said to be differentiable at }x=c\in(a,b),\ iff$
$\displaystyle \lim_{x\to c}\frac{f(x)-f(c)}{x-c}\ \text{exists finitely}$
$\displaystyle \Longleftrightarrow\ \lim_{x\to c^{-}}\frac{f(x)-f(c)}{x-c}=\lim_{x\to c^{+}}\frac{f(x)-f(c)}{x-c}$
$\displaystyle \Longleftrightarrow\ \lim_{h\to 0}\frac{f(c-h)-f(c)}{-h}=\lim_{h\to 0}\frac{f(c+h)-f(c)}{h} \\ \\ \Longleftrightarrow\ (\text{LHD at }x=c)=(\text{RHD at }x=c)$

$\displaystyle 2.\ \text{A function is said to be differentiable, if it is differentiable at every point in its domain.}$

$\displaystyle 3.\ \text{Every differentiable function is continuous but, the converse is not necessarily true.}$

$\displaystyle 4.\ \text{Following are some results on differentiability:}$
$\displaystyle (i)\ \text{Every polynomial function is differentiable at each }x\in R.$
$\displaystyle (ii)\ \text{The exponential function }a^{x},\ a>0,\ a\neq 1\text{ is differentiable at each }x\in R.$
$\displaystyle (iii)\ \text{Every constant function is differentiable at each }x\in R.$
$\displaystyle (iv)\ \text{The logarithmic function is differentiable at each point in its domain.}$
$\displaystyle (v)\ \text{Trigonometric and inverse-trigonometric functions are differentiable in their respective domains.}$
$\displaystyle (vi)\ \text{The sum, difference, product and quotient of two differentiable functions is differentiable.}$
$\displaystyle (vii)\ \text{The composition of differentiable function is a differentiable function.}$
$\displaystyle (viii)\ \text{If a function }f(x)\text{ is differentiable at every point in its domain, then}$
$\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\ \text{or,}\ \lim_{h\to 0}\frac{f(x-h)-f(x)}{-h}$
$\displaystyle \text{is called the derivative or differentiation of }f\text{ at }x\text{ and is denoted by }f'(x)\text{ or }\frac{d}{dx}(f(x)).$