Derivative Formula

A derivative helps us to know the changing relationship between two variables. Consider the independent variable 'x' and the dependent variable 'y'. The change in the value of the dependent variable with respect to the change in the value of the independent variable expression can be found using the derivative formula. Mathematically, the derivative formula is helpful to find the slope of a line, to find the slope of a curve, and to find the change in one measurement with respect to another measurement. In this section, we will learn more about the derivative formula and solve a few examples.

What is Derivative Formula?

The derivative formula is one of the basic concepts used in calculus and the process of finding a derivative is known as differentiation. The derivative formula is defined for a variable 'x' having an exponent 'n'. The exponent 'n' can be an integer or a rational fraction. Hence, the formula to calculate the derivative is:

\(\dfrac{d}{dx}.x^n = n.x^{n - 1}\)

Rules of Derivative Formula

There are some basic derivative formulas i.e. a set of derivative formulas that are used at different levels and aspects. The below image has the rules.

Derivation of Derivative Formula

Let f(x) is a function whose domain contains an open interval about some point \(x_0\). 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

Listed below are a few more important derivative formulas used in different fields of mathematics like calculus, trigonometry, etc. The differentiation of Trigonometric functions uses various derivative formulas listed here. All the derivative formulas are derived from the differentiation of the first principle.

Derivative Formulas of Elementary Functions

• \(\dfrac{d}{dx}\).xⁿ = n. x^n-1
• \(\dfrac{d}{dx}.k\) = 0, where k is a constant
• \(\dfrac{d}{dx}\).e^x = e^x
• \(\dfrac{d}{dx}\).a^x = a^x. log\(_e\) .a , where a > 0, a ≠ 1
• \(\dfrac{d}{dx}\).logx = 1/x, x > 0
• \(\dfrac{d}{dx}\). log\(_a\) e = 1/x log\(_a\) e
• \(\dfrac{d}{dx}\).√x =1/(2 √x)

Derivative Formulas of Trigonometric Functions

• \(\dfrac{d}{dx}\).sin x= cos x
• \(\dfrac{d}{dx}\).cosx= -sin x
• \(\dfrac{d}{dx}\).tan x = sec² x , x ≠ (2n+1) π/2 , n ∈ I
• \(\dfrac{d}{dx}\). cot x = - cosec² x, x ≠ nπ, n ∈ I
• \(\dfrac{d}{dx}\). sec x = sec x tan x, x ≠ (2n+1) π/2 , n ∈ I
• \(\dfrac{d}{dx}\).cosec x = - cosec x cot x, x ≠ nπ, n ∈ I

Derivative Formulas of Hyperbolic Functions

• \(\dfrac{d}{dx}\). sinhx = coshx
• \(\dfrac{d}{dx}\). coshx = sin hx
• \(\dfrac{d}{dx}\). tan hx = sec h²x
• \(\dfrac{d}{dx}\). cot hx = -cosec h²x
• \(\dfrac{d}{dx}\). sec hx = -sech hx tan hx
• \(\dfrac{d}{dx}\). cosec hx = -cosec hx cot hx

Differentiation of Inverse Trigonometric Functions

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

Differentiation of Inverse Hyperbolic Functions

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

Examples Using Derivative Formula

Example 1: Find the derivative of x⁷ using the derivative formula.

Solution:

Using the derivative formula, \(\dfrac{d}{dx}.x^n = n.x^{n - 1}\)

d/dx .x⁷ = 7.x⁶ = 7x⁶

Therefore, d/dx. x^7=7x⁶

Example 2: Differentiate 1/√x using the derivative formula.

Solution:

The derivative of 1/√x can be found using the formula \( \dfrac{d}{dx}.x^n = n.x^{n - 1} \).

\(\begin{align} \dfrac{d}{dx} . \dfrac{1}{ \sqrt x} &=\dfrac{d}{dx} . x ^\frac{-1}{2} \\&= \dfrac{-1}{2} .x^{\dfrac{-1}{2} - 1} \\ &=\dfrac{-1}{2} .x^\dfrac{-3}{2} \\ &= \dfrac{-1}{2x \sqrt x} \end{align} \)

Therefore, \( \dfrac{d}{dx} . \dfrac{1}{ \sqrt x} = \dfrac{-1}{2x \sqrt {x}} \).

Example 3: What is d/dx = Cos² x, find it by using the derivative formula.

Solution:

Let us assume t = Cosx, then dy/dx = t²

dt/dx = -sin x

Using the derivative formula, we have \(\dfrac{d}{dx}.x^n = n.x^{n - 1} \)

Here n = 2

\(\dfrac{d}{dx}\).t² = 2.t^{2 - 1}

= 2 t

By the chain rule, we have dy/dx = dy/dt . dt/dx

dy/dx = 2t . -sin x

= -2 (cos x) (sin x)

= - sin 2x

Therefore, d/dx = Cos² x is = -sin 2x