In mathematics, we use to study the term “Derivative”. It is a very important topic in both pre-calculus and calculus. The term “derivative” plays a very important role in our daily life. With the help of a derivative, we can check the velocity of a moving object. Differentiation is the process of taking the derivative.
Table of Contents
This term was introduced by Newton and Leibniz, almost 300 years ago there is a very vast history of this topic. The term “derivative” has many applications in our daily life.
In this article, we will study the basic definition of differentiation, types of differentiation i.e., implicit differentiation, explicit differentiation, partial differentiation, and the examples related to them.
What is differentiation?
“The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) concerning a change in its argument (input value)”.
Notations of differentiation
Many famous mathematicians worked on the concept of derivatives and gave their theories. Some of them introduced the notations and representations of the derivatives.
1. Newton’s notation:
Newton gave the dot notation for the derivatives i.e. Let y be any function then its derivative is given by:
ẏ, ÿ
ẏ Represents the 1st derivative.
ÿ Represents the 2nd derivative.
2. Lagrange’s notation:
Lagrange’s notation is one of the modern notations. Lagrange gave the prime I notation i.e. Let y be any function then its derivative is given by:
yI,yII
yI represents the 1st derivative.
yII represents the 2nd derivative.
3. Leibniz’s notation:
The most famous notation of the derivative dy/dx was introduced by Leibniz. He gave this notation in 1675. Let y be any function then its derivative is given by: dy/dx, d2y/dx2
dy/dx represents the 1st derivative.
d2y/dx2 represents the 2nd derivative.
Types of differentiation:
As we have discussed that the process of taking derivatives is known as differentiation it has three types.
- Explicit differentiation
- Implicit differentiation
- Partial differentiation
1. Explicit differentiation:
Explicit differentiation is the most common type of differentiation. In this type, there are two types of variables, dependent and independent. The dependent variable is always on the left side of the equation and the independent variable is always on the right side of the equation e.g., y = 2x + 1
2. Implicit differentiation:
Implicit differentiation is a type of differentiation in which both variables are on the same side. It is a bit complex differentiation as compared to explicit differentiation. In this type, the variables are not explicitly defined e.g., x2y + 5x + 3y = 9
3. Partial differentiation:
In this type of differentiation, we differentiate the function concerning a variable say “x” and all the other variables included in the function will be dealt as constants. The notation for partial differentiation is slightly different from all notations discussed above in the article.
Notation: ∂f / ∂x or f’(x) or ∂xf these three notations can be used for the 1st order partial derivative.
Examples of differentiation:
Example 1: (For explicit differentiation)
Differentiate f(x) = x / (x2 + 1) w.r.t “x”
Solution:
Let y = f(x)
So, y = x / (x2 + 1)
Step 1: Apply derivative on both sides.
d / dx(y) = d / dx {x / (x2 + 1)} ∴ (d/dx(y) = dy/dx)
dy / dx = d / dx {x / (x2 + 1)}
Step 2: Apply quotient rule i.e. f’ (a) = {v.du / dx – u.dv / dx} / v2
dy / dx = {(x2 + 1) * d/dx (x) – x * d/dx (x2 + 1)} / (x2 + 1)2 ∴ (d/dx(x) = 1)
dy / dx = {(x2 + 1) – x * d/dx (x2 + 1)} / (x2 + 1)2
Applying the sum rule on (x2 + 1)
dy / dx = {(x2 + 1) – x * d/dx (x2) + d/dx (1)} / (x2 + 1)2 ∴ d/dx (1) = 0, d/dx (x2) = 2x
dy / dx = {(x2 + 1) – x * 2x + 0} / (x2 + 1)2
Step 3: Simplify.
dy / dx = {(x2 + 1) – x * 2x} / (x2 + 1)2
dy / dx = {(x2 + 1) – 2x2} / (x2 + 1)2
dy / dx = {x2 + 1 – 2x2} / (x2 + 1)2
dy / dx = {-x2 + 1} / (x2 + 1)2
dy / dx = {1 – x2} / (x2 + 1)2
A derivative calculator can solve the above problem with a single click to avoid the larger calculations.
Example 2: (For Implicit Differentiation)
Solve x2y + 5x + 3y = 9 using rules of differentiation.
Solution:
Step 1: Apply the derivative on both sides.
d/dx (x2y + 5x + 3y) = d/dx (9)
Step 2: Separate the variables using the sum rule and apply the derivative separately.
d/dx (x2y) + d/dx (5x) + d/dx (3y) = d/dx (9)
Step 3: Separating the constants using the constant multiplication rule.
d/dx (x2y) + 5 d/dx (x) + 3 d/dx (y) = d/dx (9) ∴ (d/dx(x) = 1)
Step 4: Simplify.
Applying the product rule on “x2y”
d/dx (x2) *y + (x2) *d/dx (y) + 5 d/dx (x)+3 d/dx (y) = d/dx (9) ∴ (d/dx(y) = dy/dx), d/dx (x2) = 2x
2x * y + x2 dy/dx + 5 d/dx (x) + 3 d/dx (y) = d/dx (9)
2xy + x2 dy/dx + 5 d/dx (x) + 3 d/dx (y) = d/dx (9) ∴ (d/dx(y) = dy/dx), d/dx(x) = 1
2xy + x2 dy/dx + 5(1) + 3 dy/dx = d/dx (9)
As “9” is a constant and the derivative of any constant is always equal to zero.
2xy + x2 dy/dx + 5 + 3 dy/dx = 0
Separating variables.
3 dy/dx + x2 dy/dx = – 2xy – 5
3 dy/dx + x2 dy/dx = – (2xy + 5)
Taking dy/dx common on the L.H.S
dy/dx (x2 + 3) = – (2xy + 5)
dy/dx = – (2xy + 5)/(x2 + 3)
Final Words
In this article, we have studied the basic definition of differentiation, its notation, and its types. With time different mathematicians and physicists gave their theories and concepts on this topic. In the notation section, we learned the different notations of differentiation.
