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Differentiation is a basic mathematical concept that deals with the rate at which a function changes. Moreover, it is a mathematical process used to find the derivative of a function. This represents the tangent line’s slope to the function’s graph at any point. Notably, the Differentiation Formula is important for different applications in science, engineering and economics. Read on to learn more in detail about the Differentiation Formulas, Chart, the 7 Rules of Differentiation, Differentiation Formulas for Trigonometric Functions, Differentiation Formulas for Inverse Trigonometric Functions and more.
Who Invented Differentiation?
Differentiation as a part of Calculus was independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Additionally, Newton’s work concentrated on the motion of objects and the forces acting upon them. On the other hand, Leibniz developed much of the notation used in calculus today. However, their united steps laid the groundwork for everyday calculus.
Also Read: Difference Between Differentiation and Integration
Why is Differentiation Used?
Moreover, Differentiation is used for many purposes which include:
- Finding the rate of change: Differentiation allows us to find the rate of change of a function at a specific point. This is useful in many applications, such as optimisation problems and analysing the behaviour of functions.
- Solving optimisation problems: Differentiation is used to find the maximum or minimum values of a function. Hence, this is important in optimisation problems in fields like economics, engineering, and business.
- Analysing the behaviour of functions: Differentiation also helps understand the behaviour of functions. This includes their increasing or decreasing nature, concavity, and points of inflexion.
Also Read: Integration Formulas: Examples and Solutions
Differentiation Formula Chart
In addition, here is a chart of common Differentiation Formulas:
| Function | Derivative |
| x^n | nx^(n-1) |
| e^x | e^x |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec^2(x) |
What are the 7 Rules of Differentiation?
Here are the 7 Rules of Differentiation:
- Power Rule
The derivative of x^n is nx^(n-1)
The Formula:
- Sum Rule
The derivative of the sum of two functions is the sum of their derivatives.
The Formula:
- Product Rule
The derivative of the product of two functions is given by the product of the first function’s derivative and the second function, plus the product of the first function and the second function’s derivative.
The Formula:
- Quotient Rule
The derivative of the quotient of two functions is given by the denominator function times the derivative of the numerator function, minus the numerator function times the derivative of the denominator function, all divided by the square of the denominator function.
The Formula:
- Chain Rule
The derivative of a composite function is the product of the derivative of the inner function and the derivative of the outer function.
The Formula:
- Constant Rule
The Formula for the Constant Rule is:
here, k is a constant
- Derivative of a Constant
The Formula for the Derivative of a Constant is as follows:
here, a is a constant
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Differentiation Formulas for Trigonometric Functions
Furthermore, here are the 6 basic Trigonometric Functions:
- Derivative of sin(x):
- Derivative of cos(x):
- Derivative of tan(x):
- Derivative of cot(x):
- Derivative of sec(x):
- Derivative of csc(x):
Also Read: 10 Properties of Determinants: Formulas and Examples
Differentiation Formulas for Inverse Trigonometric Functions
In addition, here are the Differentiation Formulas for Inverse Trigonometric Functions:
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Differentiation Formula for Division
The Differentiation Formula for division is:
d / dx [ f (x) / g (x)] = [ g (x) f’ (x) – f (x) g’ (x) ] / [ g (x) ]^2
where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.
For Example:
Say that you have the function y = (x^2 + 1) / (x – 2). You want to find the derivative dy/dx.
Using the Differentiation formula for division:
d / dx [ f (x) / g (x)] = [ g (x) f’ (x) – f (x) g’ (x) ] / [ g (x) ]^2
Where,
f(x) = x^2 + 1
g(x) = x – 2
Step 1: Find the derivatives of f(x) and g(x).
f'(x) = d / dx (x^2 + 1) = 2x
g'(x) = d / dx (x – 2) = 1
Step 2: Substitute the functions and their derivatives into the formula.
dy / dx = [ (x – 2) (2x) – (x^2 + 1) (1) ] / [ (x – 2)^2]
= [2x^2 – 2x – 2x^2 – 1] / [(x – 2)^2]
= [-2x^2 – 2x – 1] / [(x – 2)^2]
Step 3: Simplify the expression.
dy/dx = -2x^2 – 2x – 1 / (x – 2)^2
Therefore, the Derivative of y = (x^2 + 1) / (x – 2) is dy / dx = -2x^2 – 2x – 1 / (x – 2)^2.
Also Read: What is the Difference Between Dot Product and Cross Product?
How do you Calculate dy dx?
To calculate dy/dx, you need to use the Differentiation formulas and rules. Here is the usual approach:
- Identify the function y = f(x).
- Apply the appropriate Differentiation formula or rule based on the function’s structure.
- Simplify the resulting expression to obtain dy/dx.
For Example:
To calculate dy/dx, you need to consider an example where you have the function y = 3x^2 + 2x – 5.
To find dy/dx for this function, you need to Differentiate it with x using the power rule and the sum rule of differentiation.
Given function: y = 3x^2 + 2x – 5
- Find dy/dx for each term: For the term 3x^2: d/dx (3x^2) = 3 * 2x = 6x
- Combine the derivatives of each term using the sum rule:
dy / dx = d / dx (3x^2) + d / dx (2x) + d / dx (-5)
= 6x + 2 + 0
= 6x + 2
Therefore, the derivative of y = 3x^2 + 2x – 5 with x is dy / dx = 6x + 2. Thus, this result represents the rate of change of the function y with x at any given point.
Also Read: Surface Area of a Cuboid: Definition, Derivation and Formula
What is the Differentiation of 2x?
The differentiation of 2x is:
d / dx (2x) = 2 ✕ d / dx (x) = 2 ✕ 1 = 2
where you use the Constant rule and the Power rule.
Also Read: Trigonometry Formulas, Examples and Solutions!
How to Differentiate a Sum?
To differentiate a Sum, you use the Sum rule. The derivative of a Sum of functions is the Sum of their derivatives.
For example, if y = f(x) + g(x), then:
dy / dx = d / dx [ f (x) + g (x) ] = d / dx [ f (x) ] + d /d x [ g(x) ]
Thus, you Differentiate each term separately and then add the results together.
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