

What is a Derivative?
Derivatives are important concepts of Mathematics. Derivatives are basic to the different solutions to the problems of calculus and differential equations. Generally, scientists observe a dynamic system to get the rate of change of some variable of interest, including this information into some differential equation and use integration methods to obtain functions that can be used to estimate the behaviour of the original system in different conditions. Let us now discuss what is derivative in Mathematics?
Derivatives in Mathematics is the rate of change of a function in terms of a variable. The rate of change of a function in derivatives can be estimated by calculating the ratio of the change of the function
As you have learned what is derivative in Mathematics, let us now discuss how to define derivatives and derivative rules that can be used to calculate many derivatives.
Define Derivatives
Let
The derivative of a function
The derivative of a function y in Leibniz’s form is expressed as:
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Derivative Rules
Below are some of the derivative rules that can be used to calculate differentiation questions.
The Constant Rule
Let
The Constant Function Rule
Let
The Power Rule
Let
The Product Rule
Let
The Chain Rule
The derivative of the function
The product rule in Leibniz's notation is represented as
The Rule
Let us consider
The Sum and Difference Rule
Let
Recall that for an arbitrary function
The sum rule states that:
The difference rule states that:
The Quotient Rule
Let
Fun Facts
Gottfried Wilhelm Leibniz introduced the symbols
and in 1675. The symbols are commonly used when the equation is examined as a functional relation between dependent and independent variables.The first derivative is represented by
or , and was once considered as an infinitesimal quotient.
Solved Examples
1. Evaluate
Solution:
Let
Then,
As we know
Accordingly,
2. Differentiate the function
Solution:
3. Find the derivative of the following function:
Solution:
We have
FAQs on Derivative Rules
Q1. Define Chain Rule.
Ans. The chain rule states that for a given composite function f(g(x)),its derivative is the derivative of outer function (leaving the inner function constant and in order) is multiplied by the derivative of the inner function. The chain rule formula is expressed as:
d/dx f(g(x)) = f'(g(x)) x g'(x)
Q2. What are the Rules for Differentiation?
Ans. The different rules for differentiation are as follows:
The general rule for differentiation is expressed as:
d/dy [yn]= ny{n-1}, where n ∈ R, and n ≠ 0.
The derivative of the constant is always equivalent to zero.
d/dx [y] = 0
The derivative of a sum is always equal to the addition of derivatives.
d/dx [a(x) + b(x)] = d/dx [a(x)] + d/dx[b(x)]
The derivative of a sum is always equal to the subtraction of derivatives.
d/dx [a(x) - b(x)] = d/dx [a(x)] - d/dx [b(x)]
The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
d/dx [a.b(x)] = a d/dx [b(x)]
Q3. When to Use the Differentiation Rule?
Ans. The differentiation rules can be applied if the given question does not specify how we should determine the derivative of a function.

















