Derivative Formula

Derivatives are the fundamental tool used in calculus. The derivative measures the steepness of the graph of a given function at some particular point on the graph. Thus, the derivative is also measured as the slope. It means it is a ratio of change in the value of the function to change in the independent variable. For example, if the independent variable is time, we often think of this ratio as a rate of change like velocity.

In this article, we are going to discuss what is a derivative in math, differential calculus formulas, basic differentiation formulas.


What is a Derivative?

Let’s know what is a derivative. 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. The inverse operation for differentiation is known as integration.In this article, we will discuss the derivative formula with examples.


What is the Meaning of the Derivative?

Here's what is the meaning of the derivative. The derivative of a function at a given point characterizes the rate of change of the function at that point. We can estimate the rate of change by doing the calculation of the ratio of change of the function Δy with respect to the change of the independent variable Δx.

In the definition of the derivative, the limit value of this ratio is considered as Δx → 0. Let us have a more rigorous formulation.


How to Find Derivative of Function?

Now how to find derivatives of the function(any) 

Suppose y = f(x) we generally use the slope formula:

The slope is equal to Change in Y/ Change in X which can be written as Δy/Δx 

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And (from the diagram given above) we see that:


Value of variable x changes from

x

to

x+Δx

Value of variable y changes from

f(x)

to

f(x+Δx)


If f is a real-valued function and ‘a’ is any point in its domain for which f is defined then f(x) is said to be differentiable at the point x=a if the derivative at a point of the function that is f'(a) exists at every point in its domain, it is given by

f’(a) = \[\frac{f(x_{0} + Δx) - f(x_{0})}{Δx}\]      

Given that the limit given above exists and that f’(a) represents the derivative at a point a of the function f(x). This is known to be the first principle of the derivative.

The domain of f’(a) can be defined by the existence of its limits. The derivative is denoted as \[\frac{d}{dx}\] f(x) = D(f(x))

 Let y = f(x) then the derivative of the function f(x) can be given as,

\[\frac{d}{dx}\] f(x) at a or  \[\frac{df}{dx}\] at a

This is known as a derivative of y with respect to x.

Also, the derivative of a function f in x at x = a is given as:

Derivative at a point of a function f(x) signifies the rate of change of the function f(x) with respect to x at a point lying in its domain. For any given function to be differentiable at any point suppose x = a in its domain, then it must be continuous at that particular given point but vice-versa is not always true.

This is how to find derivatives of a function.

 

Steps to Find Derivatives of a Function:

The steps to find the derivative of a function f(x) at point x\[_{0}\] are as follows:

  • Form the difference quotient \[\frac{f(x_{0} + Δx) - f(x_{0})}{Δx}\]     

  • Simplify the quotient, canceling Δx if possible;

  • Find the derivatives in Mathematics:

Consider the functions f and g to be two real-valued functions such that the differentiation of these functions is defined in a common domain, then let’s discuss the operations that can be performed on the functions,

 

1) Sum of Derivatives:

The Sum of derivatives of the function f and function is always equal to the derivative of their sum. \[\frac{d}{dx}\] [f(x)+g(x)] = \[\frac{d}{dx}\] f(x) +\[\frac{d}{dx}\] g(x)  Now, let us assume that f(x) = u and g(x) = v, then (u+v)’ = u’+v’ 

2) Difference of Derivatives:

The difference of derivatives of the function f and of the function g is equal to the derivative of the difference of the functions f and g.

\[\frac{d}{dx}\] [f(x)-g(x)] = \[\frac{d}{dx}\] f(x) -\[\frac{d}{dx}\] g(x) Now, let us assume that f(x) = u and g(x) = v, then (u-v)’ = u’-v’ 

3) Chain Rule Differentiation:

Here are the two functions f(x) and g(x), the chain rule formula is,

( f∘g )( x ) equals f ′ ( g( x ) )·g′( x )

4) Derivative of the Product of Two Functions: 

The derivative of the product of two functions can be solved by the product rule,

\[\frac{d}{dx}\] [f(x).g(x)] = g(x) \[\frac{d}{dx}\] f(x) +f(x) \[\frac{d}{dx}\] g(x) Now, let us assume that f(x) = u and g(x) = v, then (uv)’ = u’v+v’u 

This rule is also known as the LEIBNITZ rule for differentiating the product of any two functions.

5) Derivative of the Quotient of Two Functions:  

The derivative of the quotient of two functions can be solved by the quotient rule,

Remember that the value of the denominator should not be equal to 0. 

\[\frac{d}{dx}\] [f(x)/g(x)] =( g(x). \[\frac{df(x)}{dx}\] -f(x).\[\frac{d}{dx}\]g(x))/(g(x))²


Solved Examples on Derivative Formula:

Question 1: What is \[\frac{d}{dx}\](x³) ? Evaluate and give a solution.

Solution: We apply the formula,

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

Here, in the question n=3 so the answer is equal to 3x².


Question 2: What is the derivative of f(x) = 25 ?

Solution: Since the given function is constant, So its derivative will be zero.

i.e. f’(x) = 0

FAQ (Frequently Asked Questions)

Question 1: What is the Derivative Formula? Why Do We Calculate Derivatives?

Answer: The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x.

The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. (That means that it is a ratio of change in the value of the function to change in the independent variable.)

Question 2: What Does Derivative Measure? What is Differentiation?

Answer: The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. (That means that it is a ratio of change in the value of the function to change in the independent variable.) 

Differentiation allows us to find rates of change. For example, it allows us to find the rate of change of velocity with respect to time (which is acceleration). It also allows us to find the rate of change of x with respect to y, which on a graph of y against x is the gradient of the curve.