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) = a^{x}, a > 0 and a≠1  Domain = R
Range = (0, ∞) 

Logarithmic function  f(x) = log_{a} 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^{1}x  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:
 Find the derivative of f given by f(x) = tan^{–1}x assuming it exists.
Solution:
Let y = tan–1 x.
Then, x = tan y.
By differentiating both sides w.r.t. x, we get
1 = Sec^{2} y \(\frac{dy}{dx}\)
it implies:
\(\frac{dy}{dx}\) = \(\frac{1}{{sec}^{y}}\) = \(\frac{1}{1 + {tan}^{2}y}\) = \(\frac{1}{1 + tan({tan}^{1}x)^{2}y}\) = \(\frac{1}{1 + {x}^{2}}\)
 Differentiate e^{–x}r.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 
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