Differentiability of a function:

Differentiability applies to a function whose derivative exists at each point in its domain.

Actually, differentiability at a point is defined as: suppose f is a real function and c is a point in its domain. The derivative of f at c is defined by
\(\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\)

Differentiability in interval:

For open interval:
We can say a function f(x) is to be differentiable in an interval (a, b), if and only if f(x) is differentiable at each and every point of this interval (a, b). {As, () implies open interval}.

For closed interval:
We can say a function f(x) to be differentiable in a closed interval [a, b], if f(x) is differentiable in open interval (a, b), and also f(x) is differentiable at x = a from right hand limit and differentiable at x = b from left hand limit.

Graph of differentiable function:
when we draw the graph of a differentiable function we must notice that at each point in its domain there is a tangent which is relatively smooth and doesn’t contain any bends, breaks.

Facts on relation between continuity and differentiability:

• If at any point x = a, a function f(x) is differentiable then f(x) must be continuous at x = a but the converse may not be true.
• The best thing about differentiability is that the sum, difference, product and quotient of any two differentiable functions is always differentiable.
  If a function f(x) is continuous at x = a, then it is not necessarily differentiable at x = a.

Differentiable functions domain and range:

Functions	Curve Equation	Domain & Range	Continuity & differentiability
Identity function	f(x) = x	Domain = R Range = (-∞,∞)	Always continuous and differentiable in their domain.
Exponential function	f(x) = ax, a > 0 and a≠1	Domain = R Range = (0, ∞)
Logarithmic function	f(x) = loga x, x, a > 0 and a ≠ 1	Domain = (0, ∞) Range = R
Root function	f(x) = \(\sqrt{x}\)	Domain = [0, ∞) Range = [0, ∞)	Continuous and differentiable in (0, ∞).
Greatest Integer Function	f(x) = [x]	Domain = R Range = I	Other than integral value it is continuous and differentiable
Least integer function	f(x) = (x)	Domain = R Range = I
Fractional part function	f(x) = {x} = x – [x]	Domain = R Range = [0, 1)
Signum function	f(x) = \(\frac{|x|}{x}\) = -1,    x < 0 = 0,     x = 0 =1,     x > 0	Domain = R Range = { -1, 0, 1}	Continuous and differtentiable everywhere except at x = 0
Constant function	f(x) = c	Domain = R Range = {c}, where c is constant.
Polynomial function	F(x) = ax + b	Domain = R Range = R	Continuous and differentiable everywhere.
Sine function	Y = sin x	Domain = R Range = R	Continuous and differentiable in their domain.
Cosine function	Y = cos x	Domain = R Range = R
Tangent function	Y = tan x	Domain = R Range = R
Cosecant function	Y = cosec x	Domain = R Range = R
Secant function	Y = sec x	Domain = R Range = R
Cotangent function	Y = cot x	Domain = R Range = R
Arc sine function	Y = sin-1 x	Domain = R Range = R	Continuous and differentiable in their domain.
Arc cosine function	Y = cos-1x	Domain = R Range = R
Arc tangent function	Y = tan-1 x	Domain = R Range = R
Arc cosecant function	Y = cosec-1 x	Domain = R Range = R
Arc secant function	Y = sec-1 x	Domain = R Range = R
Arc cotangent function	Y = cot-1 x	Domain = R Range = R

Differentiability examples:

1. Find the derivative of f given by f(x) = tan^–1x assuming it exists.
   Solution:
   Let y = tan–1 x.
   Then, x = tan y.

By differentiating both sides w.r.t. x, we get
1 = Sec² y \(\frac{dy}{dx}\)
it implies:
\(\frac{dy}{dx}\) = \(\frac{1}{{sec}^{y}}\) = \(\frac{1}{1 + {tan}^{2}y}\) = \(\frac{1}{1 + tan({tan}^{-1}x)^{2}y}\) = \(\frac{1}{1 + {x}^{2}}\)

2. Differentiate e^–xr.t. x.
   Solution:
   Let y = e ^{– x}

Using chain rule, we have
\(\frac{dy}{dx}\) = e ^{– x} \(\frac{d}{dx}\) (- x) = – e ^–x

More from Calculus
Relation and Functions	Limits Formula
Continuity Rules	Derivative Formula
Integral Formula	Inverse Trigonometric function Formulas
Application of Integrals	Logarithm Formulas