Differentiation and Integration Formula
Differentiation and integration and the important branches of calculus and the differentiation and integration formula are complementary to each other. The result of the differentiation of a function on integrating gives back the original function. Differentiation is used to break down the function into parts, and integration is used to unite those parts to form the original function. Geometrically the differentiation and integration formula is used to find the slope of a curve, and the area of the curve respectively.
What is the List of Differentiation and Integration Formula?
The differentiation of a function f(x) gives f'(x), and further the integration of f'(x) gives back the original function f(x). Also sometimes the reverse process of integration is not able to generate the constant terms of the original function, and hence the constant 'C" is added to the results of the integration.
Formula 1
This basic formula of differentiation and integration is used to differentiate the simplest algebraic expression. Further, all the other higherorder algebraic expressions can be computed using this formula. Here in this formula x is the variable, n is the exponent which is a numerical value. Hee x is a variable and n is an exponent.
\[\frac{d}{dx}.x^n = nx^{n  1} \text{ and } \int x^n .dx= \dfrac{x^{n + 1}}{n + 1} + C\]
Formula 2
The differentiation of constant results in a zero, and the integration of the constant gives back the constant with a variable x. Here the constant K can be an integer, decimal, or a rational expression.
\[\frac{d}{dx}.K = 0 \text{ and } \int K.dx= Kx + C\]
Formula 3
Here we have the exponent as x and the base of the expression can be 'e' or any other numberic value represented by 'a'.
\[\frac{d}{dx}.e^x = e^x, \text{ and } \int e^x.dx= e^x + c\]
\[\frac{d}{dx}.a^x = a^x.loga \text{ and } \int a^x.dx= \dfrac{a^x}{loga} + c\]
Formula 4
The differentiation of a logarithmic function results in a unit faction with this function in the denominator, and the integration of this same unit fraction results in the logarithmic function.
\[\frac{d}{dx}.logx = \frac{1}{x} \text{ and } \int \frac{1}{x}.dx= logx+ c\]
Formula 5
The differentiation and integration of trigonometric functions are complementary to each other. Futher in the reverse process of integration the negative sign is passed on to the answer.
\(\dfrac{d}{dx}.Sinx = Cosx, \text{ and } \int Cosx.dx = Sinx + C \)
\(\frac{d}{dx}.Cosx = Sinx, \text{ and }\int Sinx.dx = Cosx + C \)
\(\frac{d}{dx}.Tanx = Sec^2x, \text{ and }\int Sec^2x.dx = Tanx + C \)
\(\frac{d}{dx}.Cotx = Cosec^2x, \text{ and } \int Cosec^2x.dx = Cotx + C \)
\(\frac{d}{dx}.Secx = Secx.Tanx, \text{ and }\int Secx.Tanx.dx = Secx + C \)
\(\frac{d}{dx}.Cosecx = Cosecx.Cotx, \text{ and }\int Cosecx.Cotx.dx = Cosecx + C \)
Formula 6
Inverse trigonometric functions has a similar differentiation and integration process as the basic trigonometric functions. Also here too for integration, the negative sign is passed on to the answer.
\(\dfrac{d}{dx}.Sin^{1}x =\dfrac{1}{\sqrt{1  x^2}}, \text{ and } \int \dfrac{1}{\sqrt{1  x^2}}.dx = Sin^{1}x + C \)
\(\dfrac{d}{dx}.Cos^{1}x =\dfrac{1}{\sqrt{1  x^2}}, \text{ and }\int \dfrac{1}{\sqrt{1  x^2}}.dx = Cos^{1}x + C \)
\(\dfrac{d}{dx}.Tan^{1}x =\dfrac{1}{1 + x^2}, \text{ and }\int \dfrac{1}{1 + x^2}.dx = Tan^{1}x + C \)
\(\dfrac{d}{dx}.Cot^{1}x =\dfrac{1}{1 + x^2}, \text{ and }\int \dfrac{1}{1 + x^2}.dx = Cot^{1}x + C \)
\(\dfrac{d}{dx}.Sec^{1}x =\dfrac{1}{x\sqrt{x^2  1}}, \text{ and } \int \dfrac{1}{x\sqrt{x^2  1}}.dx = Sec^{1}x + C \)
\(\dfrac{d}{dx}.Cosec^{1}x =\dfrac{1}{x\sqrt{x^2  1}}, \text{ and } \int \dfrac{1}{x\sqrt{x^2  1}}.dx = Cosec^{1}x + C\)
Formula 7
The following formulas of differentiation and integration can be applied for the product of two functions. Here f(x) and g(x) are the two functions.
\[\frac{d}{dx}.f(x).g(x) = g(x).\frac{d}{dx}.f(x)+ f(x).\frac{d}{dx}.g(x)\]
\[ \int f(x).g(x) = f(x).\int g(x).dx  \int[\int g(x).dx.f'(x)].dx + c \]
Let us try to more clearly understand the differentiation and integration formula, with the help of the below examples.
Solved Examples on Differentiation and Integration Formula

Example 1: Differentiate Sin^{3}x.
Solution:
d/dx.Sec^{3}x = 3Sin^{2}x. d/dx Sinx
= 3Sin^{2}x.Cosx
Answer: d/dx.Sec^{3}x = 3Sin^{2}x.Cosx 
Example 2: Find the integral of Secx(Secx + Tanx).
Solution:
\(\int \) Secx(Secx + Tanx).dx = \(\int \) (Sec^{2}x + Secx.Tanx).dx
= \(\int \)Sec^{2}x.dx + \(\int \) Secx.Tanx.dx
= Tanx + Secx + C
Answer: \(\int \) Secx(Secx + Tanx).dx = Tanx + Secx + C