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Differentiability, Theorems, Examples, Rules with Domain and Range

Differentiability for class 11

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.

graph of differentiability differentiable function

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

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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