Derivative Formula With Examples and Use

Is this page helpful?

About Derivative Formula

A derivative allows us to see how the relationship between two variables changes over time. For example, consider the independent and dependent variables 'x' and 'y.' The derivative formula can be used to calculate the change in the value of the dependent variable in relation to the change in the value of the independent variable expression. The derivative formula is used in mathematics for determining the slope of a line, the slope of a curve, and the change in one measurement with respect to another measurement. We'll study more about the derivative formula and solve a few instances in this section.

Derivation of derivative formula

Let f(x) is a function whose domain contains an open interval about some point x₀. Then the function f(x) is said to be differentiable at point (x)₀, and the derivative of f(x) at (x)₀ is represented using formula as:

f'(x)= lim_Δx→0 Δy/Δx

⇒ f'(x)= lim_Δx→0 [f((x)0 + Δx) − f(x)0)]/Δx

Derivative of the function y = f(x) can be denoted as f′(x) or y′(x).

Also, Leibniz’s notation is popular to write the derivative of the function y = f(x) as df(x)/dx i.e. dy/dx.

List of derivative formulas

A few additional essential derivative formulas used in various branches of mathematics, such as calculus and trigonometry, are listed below. The derivative formulas given here are used to differentiate trigonometric functions. The differentiation of the first principle is the source of all derivative formulas.

Formulas of elementary functions:

$\frac{d}{d x} . x^{n} = n . x^{n - 1}$
$\frac{d}{d x} . k = 0$ where k is a constant
$\frac{d}{d x} . e^{x} = e^{x}$
$\frac{d}{d x} . a^{x} = a^{x} . \log_{e} . a$ , where a > 0, a≠1
$\frac{d}{d x} . logx = \frac{1}{x}, x > 0$
$\frac{d}{d x} . \log_{a} e = \frac{1}{x} \log_{a} e$
$\frac{d}{d x} . \sqrt{x} = \frac{1}{2 \sqrt{x}}$

Formulas of trigonometric functions:

$\frac{d}{d x} . \sin x = \cos x$
$\frac{d}{d x} . \cos x = - \sin x$
$\frac{d}{d x} . tanx = \sec^{2} x, x \neq (2 n + 1) \frac{π}{2}, n \in I$
$\frac{d}{d x} . cotx = - {cosec}^{2} x, x \neq nπ, n \in I$
$\frac{d}{d x} . \sec x = \sec x \tan x, x \neq (2 n + 1) \frac{π}{2}, n \in I$
$\frac{d}{d x} . cosecx = - cosec x \cot x, x \neq nπ, n \in I$

Formulas of hyperbolic functions:

$\frac{d}{d x} . sinhx = coshx$
$\frac{d}{d x} . \cos hx = \sin hx$
$\frac{d}{d x} . \tan hx = \sec h^{2} x$
$\frac{d}{d x} . \cot hx = - {cosech}^{2} x$
$\frac{d}{d x} . \sec hx = - \sec hx \tan hx$
$\frac{d}{d x} . cosec hx = - cosec hx \cot hx$

Formulas of inverse trigonometric functions:

$\frac{d}{d x} . \sin^{- 1} x = \frac{1}{\sqrt{(1 - x^{2})}}, - 1 < x < 1$
$\frac{d}{d x} . \cos^{- 1} x = - \frac{1}{\sqrt{(1 - x^{2})}}, - 1 < x < 1$
$\frac{d}{d x} . \tan^{- 1} x = \frac{1}{(1 + x^{2})}$
$\frac{d}{d x} . \cot^{- 1} x = - \frac{1}{(1 + x^{2})}$
$\frac{d}{d x} . {cosec}^{- 1} x = - \frac{1}{| x | \sqrt{x^{2} - 1}}, | x | > 1$

Formulas of inverse hyperbolic functions:

$\frac{d}{d x} . \sinh^{- 1} x = \frac{1}{\sqrt{(x^{2} + 1})}$
$\frac{d}{d x} . \cosh^{- 1} x = - \frac{1}{\sqrt{(x^{2} - 1})}$
$\frac{d}{d x} . \tanh^{- 1} x = \frac{1}{(1 - x^{2})}$
$\frac{d}{d x} . \coth^{- 1} x = - \frac{1}{x (1 - x^{2})}$
$\frac{d}{d x} . {cosech}^{- 1} x = - \frac{1}{x \sqrt{1 + x^{2}}}$

Derivative Formula

About Derivative Formula

Derivation of derivative formula

Download pdf of Derivative Formula

NCERT Solutions All Classes