# 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:
$$\frac{dy}{dx}$$ = $$\frac{1}{{sec}^{y}}$$ = $$\frac{1}{1 + {tan}^{2}y}$$ = $$\frac{1}{1 + tan({tan}^{-1}x)^{2}y}$$ = $$\frac{1}{1 + {x}^{2}}$$

1. 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