Differentiation Formula

Differentiation, in mathematics, is the process of finding the derivative, or rate of change, of some function. The practical technique of differentiation can be followed by doing algebraic manipulations. It has many fundamental theorems and formulae for doing the differentiation of functions. In this topic, we will discuss the basic theorems and some important differentiation formula with suitable examples. Let us learn the interesting topic!

                                                                                                                                         Source:  en.wikipedia.org

Differentiation Formula

What is the differentiation?

The derivative of a function is one of the basic concepts of calculus mathematics. Together with the integral, derivative covers the central place in calculus. The process of finding the derivative is differentiation.

Differentiation allows us to determine the rates of change. For example, it allows us to determine the rate of change of velocity with respect to time to give the acceleration. It also allows us to find the rate of change of variable x with respect to y. The graph of y against x is the gradient of the curve. There are many simple rules which can be used to allow us to differentiate many functions easily.

We can estimate the rate of change by doing the calculation of the ratio of change of the function  \(\Delta y\) with respect to the change of the independent variable \(\Delta x\).

If y is some function of x then the derivative of y with respect to x is written \(\frac{dy}{dx}\) pronounced “dee y by dee x”.

Definition of the Derivative

Let f(x) is a function whose domain contains an open interval about some point \(x_0\). Then the function f(x) is said to be differentiable at point \(x_0\), and the derivative of f(x) at \(x_0\) is represented using formula as:

\(f'(x)=\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}\)

i.e. \(f'(x)=\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\)

Derivative of the function y=f(x) can be denoted as f′(x) or y′(x).

Also, Leibniz’s notation is popular to write the derivative of the function y=f(x) as

\(\frac{df(x)}{dx}\)

i.e. \(\frac{dy}{dx}\)

The steps to find the derivative of a function f(x) at the point x0 are as follows:

• Form the difference quotient \(\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\)
• Simplify the quotient, canceling \(\Delta x\) if possible;
• Find the derivative f′\((x_0)\), applying the limit to the quotient.

If this limit exists, then we can say that the function f(x) is differentiable at \(x_0\).

In the given example, we derive the derivatives of the basic elementary functions using the formal definition of a derivative. Let us assume that y = f(x) is a differentiable function at the point \(x_0\). Then the derivative of the function will be:

\(\frac{dy}{dx}_{x=x_0} = f'(x_0) = \lim_{h\rightarrow 0}\frac{f(x_0+h)-f(x_0)}{h}\)

Some Basic Derivatives 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) \(\frac{d}{dx}(u\pm v)= \frac{d}{dx}u\pm \frac{d}{dx}v\)

(5) \(\frac{d}{dx}{cu}= c\frac{d}{dx}u\)

(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

Solved Examples

Q.1: Find out the differentiation of the function \(x^5 + 2x^{-3}\).

Solution: We apply the formula

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

First, n=5 so

First term, \(5x^4\)

Then n = -3, so

\(-3 x^{-4}\)

Second term, \(-6 x^{ -4 }\)

Thus complete value of differentiation,

= \(5 x^4 – 6 x^{ -4 }\)