Differentiation and Integration Formula
Last Updated :
28 May, 2025
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
\bold{\left.\frac{d}{dx}(f(x))\right|_{x=a} =f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}}
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.
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
y = f(x) | dy/dx |
|---|
\frac{d}{dx}(x^n) | nx^{n-1} |
\frac{d}{dx} (\frac{1}{x}) | -\frac{1}{x^2} |
\frac{d}{dx} (√x) | \frac{1}{2√x} |
Derivative of Exponential Functions
y = f(x) | dy/dx |
|---|
\frac{d}{dx} (e^x) | ex |
\frac{d}{dx} (a^x) | ax logea |
Derivative of Logarithmic Functions
y = f(x) | dy/dx |
|---|
\frac{d}{dx} ( log_ex) | 1/x |
\frac{d}{dx} ( log_ax) | \frac{1}{x log_ea} |
Learn More: logarithmic differentiation
Derivative of Trigonometric Functions
y = f(x) | dy/dx |
|---|
| \frac{d}{dx} (sin x) | cos x |
| \frac{d}{dx}( cos x) | -sin x |
| \frac{d}{dx} ( tan x) | sec2x |
| \frac{d}{dx} (cot x) | -cosec2x |
| \frac{d}{dx} (sec x) | sec x.tan x |
| \frac{d}{dx} (cosec x) | -cosec x.cot x |
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.,
\bold{\frac{d}{dx}(u.v)=u\frac{dv}{dx}+v\frac{du}{dx}}
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.,
\bold{\frac{d}{dx}(\frac{u}{v} ) = \frac{v\frac{du}{dx} - u \frac{dv}{dx}}{v^2}}
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
f(x) = 2 \frac{x^2}{2} + C = x^2 + C
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
∫f'(x) | f(x) |
|---|
∫xn dx | \frac{x^n+1 }{n+1} + C |
∫\frac{1}{x}dx | logex + C |
Integration of Exponential Functions
Some commonly used formulas of Integration related to the exponential function are
∫f'(x) | f(x) |
|---|
∫exdx | ex + C |
∫ax logea dx | log_ea \frac{a^x}{log_ea} + C = a^x + C |
Integration of Trigonometric Functions
The formula for the integration of some common Trigonometric Functions are:
∫f'(x) | f(x) |
|---|
∫ cos x dx | sin x + C |
∫sin x dx | -cos x + C |
∫ cot x dx | log|sin x | + C |
∫ sec x dx | log|sec x + tan x | +C |
∫ tan x dx | -log|cos x| + C |
∫cosec x dx | log|cosec x - cot x | + C |
∫sec2 x dx | tan x + C |
∫sec x tan x dx | sec x + C |
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
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.
The formulas for Differentiation and Integration of some frequently used functions are tabulated below:
Functions | Differentiation Formula | Integration Formula |
|---|
xn | d/dx(xn) = nx(n-1) | ∫xn dx = xn+1/(n+1) + C |
|---|
1/x | d/dx(1/x) = -1/x2 | ∫(1/x)dx = loge|x| + C |
|---|
ex | d/dx(ex) = ex | ∫ex dx = ex + C |
|---|
sin x | d/dx(sin x) = cos x | ∫sin x dx = -cos x + C |
|---|
cos x | d/dx(cos x) = -sin x | ∫cos x dx = sin x + C |
|---|
tan x | d/dx(tan x) = sec2 x | ∫tan x dx = -log|cos x| + C |
|---|
cot x | d/dx(cot x) = -cosec2 x | ∫cot x dx = log|sin x| + C |
|---|
sec x | d/dx(sec x) = sec x.tan x | ∫sec x dx = log |sec x + tan x| + C |
|---|
cosec x | d/dx(cosec x) = -cosec x.cot x | ∫cosec x dx = log |cosec x - cot x| + C |
|---|
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:
Differentiation | Integration |
|---|
| Differentiation involves the Division of component functions | Integration involves the Addition of components of a function |
| Reduces the power of the function | Increases the power of the function |
| Finds the gradient or slope of the curve | Finds the area under the curve |
| Calculated for a specific point in the function's domain | Calculated for a range of points within the function's domain |
| The derivative of a function is the antiderivative of the function | The integral of a function is the antiderivative of the function |
Example 1:
Differentiate \bold{y= \frac{1}{3x+1}}with respect to x.
Solution:
Let y = \frac{1}{3x+1}
⇒ \frac{dy}{dx} = \frac{d}{dx}(\frac{1}{3x+1})
⇒ \frac{dy}{dx} = -\frac{3}{(3x+1)^2}
Example 2: Differentiate the following: i) x3 ii) \bold{\frac{1}{x^3+1}} \frac{1}{x^3+1}
Solution:
i) Let y = x3
⇒\frac{dy}{dx} = \frac{d}{dx}(x^3)
⇒\frac{dy}{dx} = 3x^2
ii) Let y = \frac{1}{x^3+1}
Using, Quotient Rule,
\frac{dy}{dx} = \frac{-3x^2}{(x^3+1)^2}
Example 3: Find of derivative of\bold{y = \frac{e^x-e^{-x}}{e^{-x}+e^x}}
Solution:
Let y = \frac{e^x-e^{-x}}{e^{-x}+e^x}
⇒ \frac{dy}{dx} =\frac{(e^{-x}+e^x)(e^x+e^{-x}) - (e^{-x}+e^x)(e^x+ e^{-x})}{(e^{-x}+e^x)^2}
⇒ \frac{dy}{dx} = \frac{e^{-2x}+e^{2x} -e^{-2x}-e^{2x}}{(e^{-x}+e^x)^2}
⇒ \frac{dy}{dx} = 0
Example 4: Differentiate \bold{y = \frac{e^x-e^{-x}}{e^{-x}+e^x}} with respect to x.
Solution:
y = \frac{e^x-e^{-x}}{e^{x}+e^{-x}}= \frac{e^{2x}-1}{e^{2x}+1}
⇒ \frac{dy}{dx}=\frac{\frac{d}{dx}(e^{2x}-1)(e^{2x}+1)-\frac{d}{dx}(e^{2x}+1)(e^{2x}-1)}{(e^{2x}+1)^2}
⇒ dy/dx = \frac{2e^{2x}(e^{2x}+1)-2e^{2x}(e^{2x}+1)}{(e^{2x}+1)^2}
⇒ dy/dx = \frac{4e^{2x}}{(e^{2x}+1)^2}
Example 5: Differentiate y = Sec2x with respect to x.
Solution:
Let y = sec2x
⇒ \frac{dy}{dx} = 2secx(secx tanx)
⇒ \frac{dy}{dx} = 2 sec2 x tan x
Example 6: Differentiate sec2x + cos2x.
Solution:
y = sec2x + cos2x
⇒ \frac{dy}{dx} = 2sinx cosx + (-2sin2x)
⇒ \frac{dy}{dx} = 2sinx cosx -2sin2x
Example 7: Integrate √x with respect to x.
Solution:
y = ∫√x dx
⇒ y = ∫x^{\frac{1}{2}} dx
⇒ y = \frac{ x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + c
⇒ y = \frac{2}{3}x^{\frac{3}{2}} + c
Example 8: Integrate the following:
(i) e2x (ii) eax
Solution:
i) y=∫e2x
⇒ y = \frac{e^{2x}}{2} + c
ii) y=∫eax
⇒ y = \frac{e^{ax}}{a} + c
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 = \frac{-cos2x}{2}+ \frac{sin2x}{2}+ c
⇒ y = \frac{1}{2}(sin2x -cos2x) + c
Example 11: Find the area bounded by the curve y = sin x between x= 0 and x = 2Ï€.
Solution:
Let y = Sinx
The graph of y = sinx is like,
Required area = Area of OABO + Area of BCDB
⇒ Required area = \int_{0}^{π}|sinx|dx + \int_{π}^{2π}|sinx|dx
⇒ Required area = \int_{0}^{π}sinx dx + \int_{π}^{2π}-sinxdx
⇒ Required area = \left[ -cosx \right]_{0}^{π} + \left[ cosx \right]_{π}^{2π}
⇒ 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.
Required Area (Shaded Area) = \int_{1}^{4}|y|dx
⇒ Required area = \int_{1}^{4}\sqrt{x}dx
⇒ Required area = \left[\frac{ x^{\frac{3}{2}}}{\frac{3}{2}} \right]
⇒ Required area = \frac{2}{3}\left[ {4}^{\frac{3}{2}} - 1 \right]
⇒ Required area = \frac{14}{3} sq. units.
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.
The area bounded by the curve is shaded portion of the graph.
Required Area = \int_{2}^{4}|x|dy
⇒ Required area = \int_{2}^{4}2\sqrt{y}dy
⇒ Required area = 2\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}} \right]_{2}^{4}
⇒ Required area = \frac{8}{3}\left[ 4 -\sqrt{2} \right] sq. units.
Question 1: Differentiate y = 5x^4-3x^2+7 with respect to x.
Question 2: Differentiate y = ln(x^2+1) with respect to x.
Question 3: Find the derivative of y = e3x sin(x) with respect to x.
Question 4: Differentiate y = x^3+2x/x^2+1 using the quotient rule.
Question 5: Find the derivative of y = cos(2x).ln(x) with respect to x.
Question 6: Integrate ∫ (4x^3-2x+1) dx.
Question 7: Evaluate the integral ∫ e2x dx.
Question 8: Find the integral of ∫ 1/x2 + 4 dx.
Question 9: Integrate ∫ (x2 + 3x +5) dx.
Question 10: Evaluate ∫ sin(3x) dx.
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Rolle's Theorem and Lagrange's Mean Value TheoremRolle's Theorem and Lagrange's Mean Value Theorem: Mean Value Theorems (MVT) are the basic theorems used in mathematics. They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges's Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value t
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Advanced DifferentiationDerivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
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Tangents and NormalsTangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the poin
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Equation of Tangents and NormalsDerivatives are used to find rate of change of a function with respect to variables. To find rate of change of function with respect to a variable differentiating it with respect to that variable is required. Rate of change of function y = f(x) with respect to x is defined by dy/dx or f'(x). For exa
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Increasing and Decreasing FunctionsIncreasing and decreasing functions refer to the behavior of a function's graph as you move from left to right along the x-axis. A function is considered increasing if for any two values x1 and x2​ such that x1 < x2 ​, the function value at x1​ is less than the function value at x2​ (i.e., f( x1)
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Increasing and Decreasing IntervalsIncreasing and decreasing intervals are the intervals of real numbers in which real-valued functions are increasing and decreasing respectively. Derivatives are a way of measuring the rate of change of a variable.Increasing and Decreasing IntervalsWhen it comes to functions and calculus, derivatives
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Concave FunctionGraphs of the functions give us a lot of information about the nature of the function, the trends, and the critical points like maxima and minima of the function. Derivatives allow us to mathematically analyze these functions and their sign can give us information about the maximum and minimum of th
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Inflection PointInflection Point describes a point where the curvature of a curve changes direction. It represents the transition from a concave to a convex shape or vice versa. Let's learn about Inflection Points in detail, including Concavity of Function and solved examples. Table of Content Inflection Point Defi
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Curve SketchingCurve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn't any sure-shot algorithm that after application spits out the graph of any desired function but
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Absolute Minima and MaximaAbsolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct
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Relative Minima and MaximaRelative maxima and minima are the points defined in any function such that at these points the value of the function is either maximum or minimum in their neighborhood. Relative maxima and minima depend on their neighborhood point and are calculated accordingly. We find the relative maxima and mini
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Integrals
Indefinite Integration
Indefinite IntegralsIntegrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
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Integration FormulasIntegration Formulas are the basic formulas used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are beneficial for finding
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Integration by Substitution MethodIntegration by substitution or u-substitution is a highly used method of finding the integration of a complex function by reducing it to a simpler function and then finding its integration. Suppose we have to find the integration of f(x) where the direct integration of f(x) is not possible. So we su
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Integration by Partial FractionsIntegration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions.If f(x) and g(x) are polynomia
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Partial Fraction ExpansionIf f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
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Integration by PartsIntegration is the calculation of an integral which is used in maths for finding many useful quantities such as areas, volumes, displacement, etc., that occur due to a collection of small data which cannot be measured singularly. Integrals are denoted using " ∫ ".Integration NotationThe concept of i
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Integration by U-substitutionFinding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
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Reverse Chain RuleIntegrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
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Trigonometric Substitution: Method, Formula and Solved ExamplesTrigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
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Integration of Trigonometric FunctionsIntegration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.Below is an example of the Integra
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Definite Integration
Definite Integral | MathematicsDefinite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits. \int_{a}^{b}F(x)dx , It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit. Note
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Computing Definite IntegralsIntegrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface
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Fundamental Theorem of Calculus | Part 1, Part 2Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for
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Finding Derivative with Fundamental Theorem of CalculusIntegrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
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Evaluating Definite IntegralsIntegration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
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Properties of Definite IntegralsProperties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as \int_{a}^{b}f(x) = F(b) − F(a)There are many properties regarding definite integral. We will discuss each property one by one with proof.Defin
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Definite Integrals of Piecewise FunctionsImagine a graph with a function drawn on it, it can be a straight line or a curve, or anything as long as it is a function. Now, this is just one function on the graph. Can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight lin
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Improper IntegralsImproper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
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Riemann SumsRiemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
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Riemann Sums in Summation NotationRiemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
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Definite Integral as the Limit of a Riemann SumDefinite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
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Trapezoidal RuleThe Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form b∫a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
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