Differentiation and Integration Formula

Integration and Differentiation are two very important concepts in calculus. These are used to study the change. Calculus has a wide variety of applications in many fields of science as well as the economy. Also, we may find calculus in finance as well as in stock market analysis. In this article, we will have some differentiation and integration formula with examples. Let us learn the interesting concept!

                                                                                                                                                                   Source:  en.wikipedia.org

Differentiation and Integration Formula

What is Differentiation?

Differentiation is the algebraic procedure of calculating the derivatives. The derivative of a function is the slope or the gradient of the given graph at any given point. The gradient of a curve at any given point is the value of the tangent drawn to that curve at the given point. For a non- linear curves, the gradient of the curve is varying at different points along the axis. Thus, it is difficult to calculate the gradient in such cases.

It is also defined as the change of a property with respect to a unit change of another property.

Let f(x) be a function of an independent variable x. Then for a small change \(\Delta\) x is caused in the independent variable x. A corresponding change \(\Delta\) f(x) is caused in the function f(x). Then the ratio:

\(\frac{ \Delta f(x)}{\Delta x}\)

is a measure of rate of change of f(x), with respect to x.

And the limit value of this ratio, as \(\Delta\) x tends to zero,

i.e. \(\lim_{\Delta x\to 0} \frac{f(x)}{\Delta x}\)

is called the first derivative of the function f(x).

What is Integration?

Integration is the process to calculate definite or indefinite integrals. For some function f(x) and a closed interval [a, b] on the real line,

the definite integral,

\(\int_{a}^{b} f(x)\;dx \)

is the area between the graph of the function, the horizontal axis, and the two vertical lines. These two lines will be at the endpoints of an interval.

When a specific interval is not given, then it is known as indefinite integral.

We will calculate the definite integral by using anti-derivatives. Therefore, integration is the reverse process of differentiation.

Remember that differentiation calculates the slope of a curve, while integration calculates the area under the curve, on the other hand, integration is the reverse process of it.

Some Basic Differentiation Formula

(1) \(\frac{d}{dx}(c)\) = 0 , c is a constant.

(2) \(\frac{d}{dx}(x)\) = 1

(3) \(\frac{d}{dx}(x^n) = nx^{n-1} \)

(4) \(\frac{d}{dx}(u\pm v)= \frac{d}{dx}u\pm \frac{d}{dx}v \)

(6) \(ddx(uv)=udvdx+vdudx \)

(7) \(\frac{d}{dx}{uv}=u\frac{d}{dx}v+v\frac{d}{dx}\)u this is Product Rule

(8) \(\frac{d}{{dx}}\left( {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \frac{{\frac{d}{{dx}}f\left( x \right)g\left( x \right) – f\left( x \right)\frac{d}{{dx}}g\left( x \right)}}{{g^2 \left( x \right)}} \)   This is Quotient Rule

Some Basic Integration Formula

(1)        \(\int 1\; dx = x+c \)

(2)        \(\int m \;dx = mx + c \)

(3)        \(\int x^n dx = \frac {x^{n+1}}{ n+1} + c \)

(4)        \(\int sinx \;dx = -cos x +c \)

(5)        \(\int cos x \;dx = sin x + c \)

(6)        \(\int sec^2 x \;dx = tan x +c \)

(7)        \(\int \frac{1}{x} \;dx = ln\; x + c \)

(8)        \(\int e^x \;dx = e^x + c \)

(9)        \(\int a^x \;dx = \frac{a^x}{ln \;a} + c \)

Solved Examples for you

Q.1: What is \(\frac{d}{dx} x^5\)?

Solution: We apply the formula

\(\frac{d}{dx}(x^n) = nx^{n-1} \)

Here n=5,  So

Solution is \(5x^4 \)