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Differentiation in Calculus | Formula, Derivative Rules and Examples

Differentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables. It is a fundamental concept in calculus used to measure the function’s per unit change in the independent variable.

In this article, we will discuss the concept of Differentiation in detail including its definition, notations, various basic rules, and many many different formulas for differentiation.

Differentiation in Calculus

Table of Content

What is Differentiation in Calculus?

Differentiation can be defined as a derivative of a function with respect to an independent variable. Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by :

f'(x)=dy / dx

Differentiation using First Principle

It states that if the function f(x) undergoes an infinitesimal small change of ‘h’ near to any point ‘x’, then the derivative of the function ( provided this limit exists ) is defined as:

f'(x)= limh → 0 [f(x+h)-f(x) ]/ h , h≠0

Differentiation Notations

If a function is denoted as y = f(x), the derivative can be indicated by the following notations.

  • D(y) or D[f(x)] is called Euler’s notation.
  • dy/dx is called Leibniz’s notation.
  • F’(x) is called Lagrange’s notation.

The meaning of differentiation is the process of determining the derivative of a function at any point say x .

Differentiation Rules

The most basic rules to find derivatives are:

Rule

Formula

Constant Rule

(d/dx) constant = 0

Power Rule

(d/dx) xn = nxn-1

Sum Difference Rule

(d/dx) [f(x) ± g(x)] = (d/dx) f(x) ± (d/dx) g(x)

Product Rule

(d/dx) [f(x). g(x)] = f'(x). g(x) + f(x). g'(x)

Quotient Rule

(d/dx) [f(x)/g(x)] = [f'(x). g(x) – f(x). g'(x)]/[g(x)]2

Chain Rule

(d/dx) [f(g(x))] = (d/dx) [f(g(x))] × (d/dx) [g(x)]

Power Rule

Power Rule states that if a variable say x is raised to some power say n then ,

(d/dx) xn = nxn-1

Product Rule

Product Rule states that if two or more functions are in product then,

(d/dx) [f(x). g(x)] = f'(x). g(x) + f(x). g'(x)

Quotient Rule

Quotient Rule states that if a function divides another function then ,

(d/dx) [f(x)/g(x)] = [f'(x). g(x) – f(x). g'(x)]/[g(x)]2

Chain Rule

Chain Rule states that if there is a composite function then ,

(d/dx) [f(g(x))] = (d/dx) [f(g(x))] × (d/dx) [g(x)]

Differentiation of Elementary Functions

Differentiation of some of the most common functions is given in the following table:

Function Type General Form Derivative Explanation
Constant Function f(x) = c f'(x) = 0 The rate of change of a constant is zero.
Linear Function f(x) = mx + b f'(x) = m The derivative of a linear function is its slope (m).
Power Function f(x) = xn f'(x) = nxn-1 Apply the power rule: bring down the exponent and subtract one from it.
Exponential Function f(x) = ex f'(x) = ex The derivative of e^x is e^x itself.
Logarithmic Function f(x) = ln(x) f'(x) = 1/x The derivative of ln(x) is 1/x.
Sine Function f(x) = sin(x) f'(x) = cos(x) The derivative of sine is cosine.
Cosine Function f(x) = cos(x) f'(x) = -sin(x) The derivative of cosine is negative sine.
Tangent Function f(x) = tan(x) f'(x) = sec2(x) The derivative of tangent is secant squared.

Differentiation Formulas

There is a list of formulas which help us in finding the derivative of a function. These are called as differentiation formulas .It can be categorized into 5 basic types:

Differentiation of Algebraic Functions

The derivative formulas for the algebraic functions are:

(d/dx)c = 0 , where c is any constant

(d/dx)x = 1

(d/dx)cx = c

(d/dx)xn = nxn-1

(d/dx)[f(x)]n = n[f(x)]n-1 .f'(x)

(d/dx) [f(x) ± g(x)] = (d/dx) f(x) ± (d/dx) g(x)

(d/dx) [f(x). g(x)] = f'(x). g(x) + f(x). g'(x)

(d/dx) [f(x)/g(x)] = [f'(x). g(x) – f(x). g'(x)]/[g(x)]2

(d/dx) [f(g(x))] = (d/dx) [f(g(x))] × (d/dx) [g(x)]

Differentiation of Trigonometric Functions

The derivative formulas for the trigonometric functions are :

(d/dx) sin x = cos x

(d/dx) cos x = -sin x

(d/dx) tan x = sec2 x

(d/dx) cot x = -cosec2x

(d/dx) sec x = sec x tan x

(d/dx) cosec x = – cosec x cot x

Learn More:

Differentiation of Logarithmic and Exponential Functions

The derivative formulas for the exponential and logarithmic functions are :

(d/dx) ex = ex

(d/dx) ax = ax ln a

(d/dx) ln x = (1/x)

(d/dx) logax= (1/x lna)

Read More:

Differentiation of Inverse Trigonometric Functions

The derivative formulas for the inverse trigonometric functions are:

(d/dx) sin-1 x = 1/[√(1 – x2)]

(d/dx) cos-1 x = 1/[√(1 – x2)]

(d/dx) tan-1 x = 1/(1 + x2)

(d/dx) cot-1 x = -1/(1 + x2)

(d/dx) sec-1 x = 1/[|x|√(x2 – 1)]

(d/dx) cosec-1 x = -1/[|x|√(x2 – 1)]

Read More:

Differentiation of Hyperbolic Functions

The derivative formulas for the hyperbolic functions are :

(d/dx) sinh x = cosh x

(d/dx) cosh x = sinh x

(d/dx) tanh x = sech2 x

(d/dx) coth x = -cosech2x

(d/dx) sech x = -sech x tanh x

(d/dx) cosech x = -cosech x coth x

Differentiation Techniques

Some techniques of differentiation include:

  • Implicit Differentiation
  • Logarithmic Differentiation
  • Parametric Differentiation

Implicit Differentiation

The method of finding the derivative of an implicit function is called Implicit Differentiation. It is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function.

The general representation of implicit differentiation can be d/dx{f(x,y)}.

df/dx = (df/dy).(dy/dx)

Logarithmic Differentiation

Logarithmic differentiation is a method used in calculus to find the derivative of complicated functions by first taking the natural logarithm of both sides of the equation, applying properties of logarithms to simplify the equation, and then differentiating implicitly using the chain rule.

Read More about Logarithmic Differentiation.

Parametric Differentiation

Parametric differentiation is a technique used to find the derivative of a function defined by parametric equations. It involves differentiating the parametric equations with respect to the parameter and then using the ratio of the derivatives to find the derivative of one variable with respect to the other.

Read More about Parametric Differentiation.

Higher Order Differentiation

Finding the derivative of a function for more than one time gives the higher drder derivative of a function.

nth Derivative = dny/(dx)n

Partial Differentiation

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

For a multivariable function, like f(x, y) = x2y, computing partial derivatives looks something like this. Partial Derivative of f(x, y) w.r.t x is given by:

∂f/∂x = ∂(x2y)/∂x = 2xy, treating y as a constant

Partial Derivative of f(x, y) w.r.t y is given by:

∂f/∂y = ∂(x2y)/∂y = x2(1), treating x as a constant

The symbol ‘∂” is called as “del“, and it is used to distinguish partial derivatives from ordinary single-variable derivatives.

Related :

Differentiation Examples

Example 1 : Differentiate f(x) = x3 – 9x2 + 4x+1 with respect to x.

Solution:

On differentiating both the sides w.r.t x, we get;

f'(x) = 3x2 -9(2)x+4x0+0 = 3x2-18x+4

Example 2 :Differentiate y = x(3x2 – 9)

Solution:

Given , y = 3x3 – 9x = f(x)

On differentiating both the sides we get,

dy/dx = 9×2 – 9 = f'(x)

Example 3 : Differentiate f(x) = x3 sin 2x

Solution:

On differentiating both the sides w.r.t x, we get;

f'(x) = 3x2(sin2x) + x3(2cos2x) = 3x2(sin2x) + 2x3(cos2x)

Example 4 : Differentiate g(x) = 4xe2x − 9x

Solution:

On differentiating both the sides w.r.t x, we get;

g'(x) = 4e2x+4x(2e2x) – 9 = 4e2x + 8xe2x – 9

Differentiation Practice Questions

Find the derivative of the following functions:

  1. f(x) = 5x3-2x2+7x-1
  2. g(x) = 1/x2 + 4x
  3. k(x) = [x2+2x+1]/[x+1]
  4. h(x) = (2x2+3x)(3x-5)
  5. p(x)=sin(3x+1)
  6. q(x)=e10x

Frequently Asked Questions on Differentiation

Define Derivative in Maths.

A derivative in mathematics represents the rate at which a function changes at a given point, essentially measuring the slope of the function’s curve at that point.

How do you Solve Differentiation Problems?

To solve differentiation problems, identify the function, apply differentiation rules (like power, product, quotient, chain rules), simplify the result, and evaluate at specific points if required.

What is an Example of Differentiation in Real Life?

An example of differentiation in real life is calculating the speed of a car at a specific time, given its distance-time graph. The derivative gives the car’s velocity at that moment.

What is Derivative of Constant Function?

The derivative of a constant function is zero, as a constant does not change and therefore has no rate of change.

What is Derivative of sin x?

The result of differentiating sinx is cos x.

What is Derivative of tan x?

The derivative of tan(x) with respect to x is sec2(x).

What is Derivative of sec x?

The derivative of sec(x) with respect to x is sec(x)tan(x).

Is Differentiation the same thing as Derivative?

Yes, derivative and differentiation are the same thing


 


Reffered: https://www.geeksforgeeks.org

Class 12

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