Differentiation Formulas

Differentiation Formulas: Differentiation allows us to analyze how a function changes over its domain. We define the process of finding the derivatives as differentiation. The derivative of any function f(x) is represented as d/dx.f(x)

In this article, we will learn about various differentiation formulas for Trigonometric Functions, Inverse Trigonometric Functions, Logarithmic Functions, etc., and their examples in detail.

Differentiation is defined as the rate of change of one quantity with respect to the other quantity. .

For any function y = f(x), if the input value changes from x to x + h then y = f(x + h) then the differentiation of the f(x) with respect to x is defined as,

dy/dx = lim_h→0 {f(x+h) – f(x)}/{(x+h) – x}

Mathematically,

dy/dx = f'(x) = lim_h→0 {f(x+h) – f(x)}/h

Read in Detail: Calculus in Maths

Differentiation formulas are used to find the differentiation of the various functions. The first principal formula states that, for any function f(x) its derivative with respect to x is,

f'(x) = lim_h→0 {f(x+h) – f(x)}/h

Read in Detail: Derivative Formulas in Calculus

The differentiation formulas for some elementary functions are:

Derivatives of the trigonometric functions are:

The differentiation formulas for the Inverse trigonometric functions are:

Let’s discuss the Differentials of Hyperbolic functions.

Various rules of finding the derivative of functions have been given below:

Differentiation of Special Functions

If we have two parametric functions x = f(t), y = g(t), where t is the parameter, then the differentiation of parametric functions is as follows,

As dy/dt = g'(t) and dx/dt = f'(t) then dy/dx is given by:

dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t)

Some functions are of the type where separating dependent variable (y) and independent variable (x) is not possible these functions are of the form f(x,y) = 0 the differentiation of these functions is not found using the normal formulas then the differentiation of these functions is found using the concept as shown in the example added below,

Example: Find the differentiation of x² + y² + 4xy = 0

Solution:

x² + y² + 4xy = 0

Differentiating with respect to x,

2x + 2ydy/dx + 4(xdy/dx + y) = 0

⇒ 2x + 4y + 2dy/dx(y + 2x) = 0

⇒ x + 2y + dy/dx(y + 2x) = 0

⇒ dy/dx(y + 2x) = -(x + 2y)

⇒ dy/dx = -(x + 2y)/(y + 2x)

Higher Order Differentiation

Higher order differentiation is nothing, but the differentiation of a function more than one time suppose we have a function y = f(x) then its differential in higher order is calculated as,

First Derivative = dy/dx = f'(x)

Second Derivative = d²y/dx² = f”(x)

Third Derivative = d³y/dx³ = f”'(x)

….
….

n^th Derivative = dⁿy/dxⁿ = f⁽ⁿ⁾(x)

This can be understood using the example added below,

Example: Find the second-order derivative of f(x) = 4x⁴ + 3x³ + 2x² + x + 1

Solution:

f(x) = 4x⁴ + 3x³ + 2x² + x + 1

Differentiating with respect to x,

f^‘(x) = 4(4x³) + 3(3x²) + 2(2x) + 1 + 0

⇒ f'(x) = 16x³ + 9x² + 4x + 1

For second-order derivative differentiating with respect to x,

f”(x) = 16(3x²) + 9(2x) + 4 + 0

⇒ f”(x) = 48x² + 18x + 4

This is the required second-order derivative.

Articles Related to Differentiation Formulas:

• Differentiation and Integration Formula
• Differential Equations
• Advanced Differentiation
• Implicit Differentiation
• Logarithmic Differentiation
• Calculus | Differential and Integral Calculus

Let’s solve some example problems on the rules of derivative.

Example 1: Find the differentiation of y = 4x³ + 7x² + 11x + 12

Solution:

Given

• y = 4x³ + 7x² + 11x + 12

Differentiating with respect to x,

dy/dx = 4(3x²) + 7(2x) + 11(1) + 0

⇒ dy/dx = 12x² + 14x + 11

This is the required differentiation

Example 2: Find the differentiation of y = cos(log x)

Solution:

Given

• y = cos(log x)

Differentiating with respect to x,

dy/dx = d/dx{cos (log x)}

⇒ dy/dx = sin (log x).{d/dx(log x)}

⇒ dy/dx = sin (log x).(1/x)

This is the required differentiation

Example 3: Find the differentiation of y = tan (3x² + 4x)

Solution:

Given

• y = tan (3x² + 4x)

Differentiating with respect to x,

dy/dx = 1/{1 + (3x² + 4x)²}² d/dx(3x² + 4x)

⇒ dy/dx = 1/{1 + (3x² + 4x)²}² (6x + 4)

⇒ dy/dx = (6x + 4)/{1 + (3x² + 4x)²}² 

This is the required differentiation

Problem 1: Find the derivative of the function f(x) = 3x² + 5x – 2.

Problem 2: Determine the derivative of [Tex]g(x) = \frac{1}{x}[/Tex].

Problem 3: Find the derivative of [Tex]h(x) = \sqrt{x^3 + 2x – 1}[/Tex].

Problem 4: Determine the derivative of [Tex]y(x) = e^{2x}[/Tex].

Problem 5: Find the derivative of [Tex]f(x) = \ln(x^2 + 3x)[/Tex].

Differentiation formulas are essential tools in calculus that help us find the rate at which things change. By using these formulas, we can determine the slope of a curve at any point, understand how one variable affects another, and solve many real-world problems involving rates of change. These formulas simplify the process of finding derivatives, making it easier to analyze and predict the behavior of various functions.

What is Differentiation?

Differentiation is defined as the rate of change of one quantity with respect to the other quantity. We represent the differentiation of y = f(x) as, dy/dx.

What is Product rule of Differentiation?

The differentiation of Product rule is,

For the function f(x) = u.v the differentiation of f(x) is,

f'(x) = ud/dx(v) + vd/dx(u)

What is Differentiation of cot x?

The differentiation of y = cot x is,

dy/dx = -cosec²x

What is the Differentiation of sec x?

The differentiation of y = sec x is,

dy/dx = sec x. tan x

What is Differentiation of log x?

The differentiation of y = log x is,

dy/dx = 1/x

What is Differentiation of tan x?

The differentiation of y = tan x is,

dy/dx = sec² x

Why are differentiation formulas important?

They are important because they help us understand the behavior of various functions. By finding the rate of change, we can solve real-world problems and make predictions.

What is a derivative?

A derivative is the result we get when we apply a differentiation formula. It tells us the slope of a curve at any given point, showing how one variable changes with another.

How do differentiation formulas simplify finding derivatives?

Differentiation formulas provide specific rules and shortcuts that make it much easier and quicker to find derivatives, rather than doing complex calculations every time.

Can you give an example of how differentiation is used in real life?

Yes, for example, in physics, differentiation can be used to find the speed of a moving object at any instant by differentiating its position with respect to time.

What does it mean to find the slope of a curve at a point?

Finding the slope of a curve at a point means calculating how steep the curve is at that specific spot. This is done using differentiation formulas to get the derivative.

How do these formulas help in understanding relationships between variables?

By using differentiation formulas, we can see how changes in one variable affect another. For example, in economics, we can understand how changes in price affect demand.

Are differentiation formulas only used in math?

No, they are used in many fields like physics, engineering, economics, and biology to solve problems and understand changes.

Is learning differentiation formulas difficult?

While it can be challenging at first, with practice, understanding and using differentiation formulas becomes easier. They are logical and follow set rules.

10. How do differentiation formulas help predict behavior?

They help predict how a function will behave in the future by showing the rate of change. For instance, in population studies, they can help predict growth rates.