Differentiation and Integration Formula

Differentiation and Integration are two mathematical operations used to find change in a function or a quantity with respect to another quantity instantaneously and over a period, respectively. Differentiation is an instantaneous rate of change, and it breaks down the function for that instant with respect to a particular quantity, while Integration is the average rate of change that causes the summation of continuous data of a function over the given period or range. Both are the inverse of each other.

In this article, we will learn about what differentiation is, what integration is, and the formulas related to Differentiation and Integration.

What is Differentiation?

Differentiation is a method to find the instantaneous rate of change of a function or curve with respect to other quantities. Mathematically, the Slope of the tangent at a point on the curve is called the Derivative of the Curve or Function, and differentiation is a method to find that derivative. In differentiation, we compute the rate at which a dependent variable 'y' changes with respect to the change in the independent variable 'x'. This rate of change is called the derivative of 'y' with respect to 'x', where y is a function of x given as y = f(x).

How to Differentiate a Function

The differentiation of a function is simply the Derivative of the Function at all differentiable points in its domain. For Example, if f(x) is differentiable at x = a in its domain. The Derivative of f(x) at x = a is given as

where h is the change in independent variable x.

For example, if we need to find the differentiation of a function f(x) = x².

Then the differentiation of the function is given as f'(x) = 2x.

Here, in the above step, we reduce the power to 1.

Differentiation Formulas

The derivative of standard functions can be found by the formulas. We will learn the Differentiation formulas of the following functions:

• Algebraic Function
• Exponential Function
• Logarithmic Function
• Trigonometric Function

Derivative of Algebraic Functions

Derivative of Exponential Functions

Derivative of Logarithmic Functions

Learn More: logarithmic differentiation

Derivative of Trigonometric Functions

The above formulas are used when the functions are present alone or when multiplied by a scalar number, but when two functions are in product form or quotient form, then we can't simply differentiate each function separately; we need to follow some rules, particularly for the product and quotient case. Hence, we will look at differentiation by parts.

Also, read Differentiation of Inverse Trigonometric Functions.

Differentiation by Parts

Differentiation of Product of Functions: Let us assume 'u' and 'v' are two functions in the product (u.v), then the Differentiation of u.v is given by the product rule i.e.,

Differentiation of Quotient of Functions: Let us assume 'u' and 'v' are two functions in the quotient (u/v), then the Differentiation of  u/v is given by the quotient rule i.e.,

What is Integration?

Integration is a method to find the average rate of change of a function. As the name suggests, Integrate means adding all the functions' points. Integration is actually the anti-derivative of a differentiating function. Differentiation and Integration are inverses of each other. We can integrate the function in two ways, one is indefinite and the other is definite. In Indefinite Integration, we get a constant C with our expression, but in Definite we can find the value of that constant C by restricting its range or limit. The Integration of a function f(x) is given as

∫f(x)dx = F(x) + C

Where, 

• f(x) is Integrand,
• dx is Integrating Agent,
• F(x) is anti-derivative of f(x), and
• C is Constant.

How to Integrate a Function

To explain we considered above result i.e. (1) - derivative of function f(x)

f'(x) = 2x 

Integrating both sides,

∫f'(x) = ∫2x 

Here,

We have to increase the power of derivative by 1 and also divide function with updated power of function, 

After that add an integral constant with it. 

Integration is called anti-derivative

Integration Formulas

To integrate various types of functions, we have different formulas for different types of functions. We will learn Integration Formulas for the following functions:

• Algebraic Function
• Exponential Function
• Trigonometric Function

Integration  of Algebraic Functions

Formulas for the Integration  of Algebraic Functions are

Integration  of Exponential Functions

Some commonly used formulas of Integration related to the exponential function are

Integration of Trigonometric Functions

The formula for the integration of some common Trigonometric Functions are:

The above formulas are used when the functions are present alone or when multiplied by a scalar number, but when two functions are in product form or quotient form, then we can't simply integrate each function separately; we need to follow some rules, particularly for the product and quotient cases. Hence, we will look at Integration by Parts.

Integration By Parts

In integration by parts, we will learn the formulas for Integration when two functions are in product or quotient form:

Integration of Product of Functions: Let us assume 'u' and 'v' are two functions in the product (u.v), then the Integration of u.v is given as

 ∫u.v dx= u∫v dx - ∫ [(du/dx) ∫vdx] dx

Integration of Quotient of Functions: Let us assume 'u' and 'v' are two functions in the product (u/v), then the Integration of u/v is given as

∫u/v = u∫(1/v) dx - ∫ [(du/dx) ∫(1/v)dx)] dx

Also, Read 

• Integration by Substitution
• Definite Integrals
• Indefinite Integrals

Area Under the Curve

Area Under the Curve refers to the region enclosed by the graph of a function and the coordinate axes, or the intersection region of two graphs. Here, we will not have a regular shape, hence we can't use regular formulas. To calculate the area in such a case, we will use the concept of Integration. We will take an elemental area dx under the curve and integrate it over the defined range x = a to x = b.

👁 Area Under the Curve

Basic Differentiation and Integration Formulas

The formulas for Differentiation and Integration of some frequently used functions are tabulated below:

Properties of Differentiation and Integration

The Properties of Differentiation and Integration are listed below:

• Property of Scalar Multiplication: Both Differentiation and Integration follow the rule of scalar multiplication, i.e.,

 d/dx{k.f(x)} = k.d/dx{f(x)} and ∫k.f(x)dx = k.∫f(x)dx 

Where k is a scalar quantity.

• Properties of Addition and Subtraction: If functions are in the Addition and Subtraction form, we need to solve Differentiation and Integration as 

d/dx{f(x) ± g(x)} = d/dx{f(x)} ± d/dx{g(x)}
and
∫{f(x) ± g(x)} dx = ∫{f(x)}  dx± ∫{g(x)} dx

• Differentiation and Integration are the inverse processes of each other, i.e., 

d/dx{f(x)} = f'(x) and ∫f'(x) dx = f(x)

• Differentiation follows the Chain Rule, i.e., if we have a function of a function, then the derivative of it is given as the derivative of the first function multiplied by the derivative of the second function, i.e.,

d/dx{f(g(x))} = f'(x).g'(x)

• Both Differentiation and Integration can be solved for a given limit.

Differentiation vs Integration

The difference between Differentiation and Integration is as follows:

Solved Examples of Differentiation and Integration Formulas

Example 1:

Differentiate with respect to x.

Solution:  

Let 

⇒  

⇒ 

Example 2: Differentiate the following: i) x³ ii)

Solution: 

i) Let y = x³

⇒\frac{dy}{dx} = \frac{d}{dx}(x^3)                                

⇒

 ii) Let

Using, Quotient Rule,                              

Example 3: Find of derivative of 

Solution:

Let 

⇒ 

⇒ 

⇒ 

Example 4: Differentiate with respect to x. 

Solution: 

⇒

⇒ 

⇒ 

Example 5: Differentiate y = Sec²x with respect to x. 

Solution: 

Let y = sec²x           

⇒ 

⇒  = 2 sec² x tan x

Example 6: Differentiate sec²x + cos2x.

Solution:

y = sec²x + cos2x                    

⇒ 

⇒ 

Example 7: Integrate √x with respect to x.

Solution:  

y = ∫√x dx

⇒ y = 

⇒ 

⇒ 

Example 8:  Integrate the following: 

(i) e^2x     (ii) e^ax

Solution:  

i) y=∫e^2x 

⇒ 

ii) y=∫e^ax

⇒  

Example 9: Integrate  sin²x+ + cos²x.

Solution:

y = ∫(sin²x + cos²x)dx          

⇒ y = ∫dx                      

⇒ y = x + c  

Example 10: Integrate sin 2x + cos 2x.

Solution: 

y = ∫(sin2x + cos2x)dx 

⇒ y = ∫sin2xdx + ∫cos2x dx

⇒ y = 

⇒ 

Example 11: Find the area bounded by the curve y = sin x between x= 0 and x = 2π.

Solution: 

Let y = Sinx 

The graph of  y = sinx is like,

👁 y = sin x

Required area = Area of OABO + Area of BCDB 

⇒ Required area 

⇒ Required area 

⇒ Required area 

⇒ Required area = -cosπ + cos0 + cos2π- cosπ

⇒ Required area = 4 sq units. 

Example 12: The area bounded by the region of the curve y² = x and the lines x = 1, x = 4, and the x-axis is :

Solution:

Let y² = x a curve region bounded by the lines x = 1 and x = 4 about x-axis. 

👁 y^2 = x

Required Area (Shaded Area)  =

⇒ Required area 

⇒ Required area 

⇒ Required area 

⇒ Required area 

Example 13: The area of the region area integrate x with respect. y and take y = 2 as the lower limit and y = 4 as the upper limit. The given curve x^2 = 4y is a parabola, which is symmetrical about the y-axis. 

Solution:  

The given curve is parabola x² = 4y which is symmetric to the y-axis. 

👁 x^2 = 4y

The area bounded by the curve is shaded portion of the graph.

Required  Area = 

⇒ Required area 

⇒ Required area 

⇒ Required area 

Practice Problems on Differentiation and Integration Formulas

Question 1: Differentiate y = with respect to x.

Question 2: Differentiate y = with respect to x.

Question 3: Find the derivative of y = e^3x sin(x) with respect to x.

Question 4: Differentiate y = using the quotient rule.

Question 5: Find the derivative of y = cos(2x).ln(x) with respect to x.

Question 6: Integrate ∫ () dx.

Question 7: Evaluate the integral ∫ e^2x dx.

Question 8: Find the integral of ∫ 1/x² + 4 dx.

Question 9: Integrate ∫ (x² + 3x +5) dx.

Question 10: Evaluate ∫ sin(3x) dx.