Derivative calculus is an important and core dimension of calculus. Differential calculus imparts a framework for us to model, explain and make anticipations about real-life problems. Calculus was first devised in the 19th century, and today it is the most critical middle domain of mathematics, which plays a vital role in the development of modern science.

Sir Isaac Newton and Gottfried autonomously expanded the idea of the derivative function, which is also sometimes termed as differential calculus in the 17th century. Although they both arrived at similar expressions for the derivative, they approached the problem differently.

Newton considered variables changing w.r.t time function, while Leibniz gave an idea of x and y variables spanning a sequence of infinitely close values. In this article, we will explore the idea of derivatives, some specific rules, its applications as well as we will solve some its examples.

What is a Differential Calculus?

The instantaneous rate of change of a function at a specific given point, say A, is called the differential of f w.r.t x. We sometimes use different notations for derivative functions:

• f’(x) (read as “f prime of x” or “f dash of x”)
• dy/dx (meaning “the derivative of y w.r.t x”

Rules of Differential Calculus:

Here we will discuss some of the essential rules of differentiation which are used in the process of the finding derivative of the functions as well as for the simplification of the problems related to differentiation.

Power Rule:

If  ‘n’ is any real number and f(x) = xⁿ, then the power rule for differentiating xⁿ is given as

f’(x) = n x^n-1 

Derivative of Constant and Linear Functions:

If f(x) = c where c  R, then f’(x) = 0
�.    

If f(x) = mx + c where m, c  R, then f’(x) = m.

i.e. Constant function results in zero on differentiation.

Derivative of a Constant Multiple of a Function:

For c  R, (cf)’(x) = c (f’(x), provided f’(x) exists.

Derivative of the Sum of Function:

Let p(x) and q(x) are differentiable functions, then

(p(x) + q(x))’ = p’(x) + q’(x). It means that the derivative of the sum of two functions is equal to the sum of the derivatives of their functions.

Derivative of the Difference of Functions:

If p(x) and q(x) are two differentiable functions, then the derivative of the difference of the functions is equal to the difference of their derivatives. Mathematically,

(p(x) – q(x))’ = m’(x) – n’(x)

Product Rule:

Let us have two differentiable functions p(x) and q(x). Then, the product rule is defined as

(p(x) * q(x))’ = p’(x) q(x) + p(x) q’(x)

Quotient Rule:

If f(x) and g(x) are differentiable functions, then the quotient rule of differentiation is defined as

(p(x)/ q(x))’ = (p’(x) q(x) – p(x) q’(x))/ q(x)²   

A differential calculator can solve the problems of differentiation according to the above laws of differential calculus.

Examples of Differential Calculus

Here we will solve some examples using differentiation rules.

Example 1:

Find the derivative of the function f(x) = x¹¹ using power rule.

Solution:

Step 1: Write down here the given function.

f(x) = x¹¹ 

step 2: Write down here the power rule of differentiation.

f’(x) = n x^n-1 

Step 3: Apply the power rule and simplify.

F(x) = 11 x¹⁰

Example  2:

Compute the derivative of the function f(x) = -x + 2 by applying constant and linear rule of differentiation.

Solution:

Step 1: Write down here the given function.

f(x) = -x + 2

Step 2: Write down here the constant and linear rule of differentiation.

If f(x) = c, then f’(x) = 0
�.    

If f(x) = mx + c, then f’(x) = m.

Step 3: Apply the constant and linear rule of differentiation.

f(x) = -1 + 0

f(x) = -1

Example 3:

Find the derivative of the function f(x) = 3x² using the constant multiple rule of differentiation.

Solution:

Step 1: Write down here the given function.

f(x) = 3x² 

Step 2: Write down here the constant multiple rule of differentiation.

(cf)’(x) = c (f’(x)

Step 3: Apply the constant multiple rule of differentiation.

f’(x) = 3 (x²)’

f’(x) = 3 (2x)

f’(x) = 6x

Example 4:

Find the derivative of the function f(x) = x⁵ – 3x² + x

Solution:

Step 1: Given function:

f(x) = x⁵ – 3x² + x

Step 2: Write down the sum and difference rule of the differentiation of function.

(p(x) + q(x))’ = p’(x) + q’(x)

(p(x) – q(x))’ = p’(x) – q’(x)

Step 3: Apply the sum and difference rule and simplify.

f(x)’ = (x⁵)’ – (3x²)’ + (x)’

f(x)’ = (5x⁴) – 3(2x) + 1  (constant multiple & power rule)

f(x)’ = 5x⁴ – 6x + 1

Example 5: 

Find the derivative of the function y(x) = 5x²(x + 3)

Solution:

Step 1: Given function:

y(x) = 5x²(x + 3)

Here let f(x) = 5x², g(x) = x + 3

Step 2: Product rule of differentiation is

(p(x) * q(x))’ = p’(x) q(x) + p(x) q’(x)

Step 3: Apply product rule and simplify.

y’(x) = (5 * 2x)(x + 3) + 5x²(1 + 0)    (using constant and linear function rule)

y’(x) = 10x(x + 3) + 5x²(1)

y’(x) = 10x² + 30x + 5x² 

y’(x) = 15x² + 30x

Example 6:

Find the derivative of the function y(x) = (x + 3)/ x²

Solution: 

Step 1: Given function:

y(x) = (x + 3)/ x² 

Here let f(x) = x + 3, g(x) = x²  

Step 2: Write down the Quotient rule of the differentiation.

(p(x)/ q(x))’ = (p’(x) q(x) – p(x) q’(x))/ q(x)² 

Step 3: Apply the quotient rule and simplify.

y’(x) = ((1 + 0)(x²) – (x +3)(2x))/ (x²)²     (using constant and linear rule)

y’(x) = ((1)(x²) – (2x² + 6x))/ x⁴ 

y’(x) = (x² – 2x² – 6x)/ x⁴ 

y’(x) = (-x² – 6x)/ x⁴ 

Summary:

In this article, we have discussed the concept of derivative calculus. We’ve elaborated the definition and the rules of derivatives. In the last section, we solved some examples to find out derivative of different functions following rules of differentiation.