## Differential calculus: Defined and explained with examples

In the modern era, mathematics become a compulsory part of various branches of education. One of the commonly used branches of mathematics is calculus. Calculus is further divided into different categories.

The two commonly used types of calculus are integral calculus and differential calculus. These two types of calculus are rather tricky and students face difficulties in solving their problems. In this post, we will discuss differential calculus along with calculations to help the students who find this topic rather tough.

## What is differential calculus?

According to Wikipedia:

The differential calculus is a subtype of calculus in which the rate of change of quantities is studied. This branch of calculus is widely used in geometry to find the sole of the tangent line.

The process of finding the derivative of the function with respect to the independent variable is known as differentiation.

It is frequently used to solve complex calculus problems with the help of rules of differentiation. It is a wide concept in mathematics and is taught in every field of science and engineering.

The differential calculus is widely used to find the derivative of one variable, two variables, and multivariable functions. For solving these variable problems there are three different types of derivatives

• Explicit differentiation
• Implicit differentiation
• Partial differentiation

Let’s further explore the subtypes of differential calculus.

### 1.   Explicit differentiation

It is the most commonly used type of differential calculus as it differentiates the one variable functions with respect to the independent variables. The single variable function is a function that involves only one variable in the function.

The function can be linear, polynomial, or constant. It is denoted by d/dx or f’(x).

### 2.   Implicit differentiation

It is the subtype of differential calculus that deals with two variable functions or implicit functions to find the dy/dx of the function. It differentiates the dependent variable with respect to the independent variable without considering it constant.

### 3.   Partial differentiation

To differentiate the multivariable functions this, type of differential calculus is used. It differentiates the multivariable function with respect to the independent variable. It differentiates one variable at a time and considers the other variables as a constant.

## How to calculate the differential calculus problems?

There are two methods for solving the problems of differential calculus.

• By using a derivative calculator
• Manually

Here is a brief introduction to both methods.

### By using a derivative calculator

As it is the era of technology. There are hundreds of online tools are available to help the students for getting the step by step solutions to their complex problems. The problems of differential calculus can be solved easily by using a derivative calculator.

How to use this calculator?

• Enter the function into the required input field.
• Select the corresponding variable “x” is selected by default.
• Enter the number of derivatives you want to calculate i.e., 1 for the first derivative, 2 for the second, and so on.
• Press the calculate button.
• The solution with steps will come below the calculate button.

### Manually

Here are a few solved examples to solve the differential calculus problems manually.

Example 1

Evaluate the differential of the given function w.r.t “t”.

p(t) = 12t3 + 2t4 – 7t2 + 5t + 12

Solution

Step-1: Take the given function, and independent variable, and apply the notation of differentiation to it.

p(t) = 12t3 + 2t4 – 7t2 + 5t + 12

independent variable = t

d/dt p(t) = d/dt [12t3 + 2t4 – 7t2 + 5t + 12]

Step-2: Now separate each function with the help of sum and difference laws and apply the notation of differential calculus.

d/dt [12t3 + 2t4 – 7t2 + 5t + 12] = d/dt [12t3] + d/dt [2t4] – d/dt [7t2] + d/dt [5t] + d/dt [12]

Step-3: Take the constant coefficients outside the derivative notation with the help of the constant function law of the differential notation.

d/dt [12t3 + 2t4 – 7t2 + 5t + 12] = 12d/dt [t3] + 2d/dt [t4] – 7d/dt [t2] + 5d/dt [t] + d/dt [12]

Step-4: Now differentiate the above expression with the help of the power rule.

d/dt [12t3 + 2t4 – 7t2 + 5t + 12] = 12 [3 t3-1] + 2 [4 t4-1] – 7 [2 t2-1] + 5 [t1-1] + [0]

= 12 [3 t2] + 2 [4 t3] – 7 [2 t1] + 5 [t0] + [0]

= 12 [3 t2] + 2 [4 t3] – 7 [2 t] + 5 [1]

= 36 [t2] + 8 [t3] – 14 [t] + 5 [1]

= 36t2 + 8t3 – 14t + 5

Example 2

Evaluate the differential of the given function w.r.t “w”.

p(w) = 3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)

Solution

Step-1: Take the given function, and independent variable, and apply the notation of differentiation to it.

p(w) = 3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)

Independent variable = w

d/dw p(w) = d/dw [3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)]

Step-2: Now separate each function with the help of sum and difference laws and apply the notation of differential calculus.

d/dw [3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)] = d/dw [3w3] + d/dw [12w4] – d/dw [7cos(w)] + d/dw [5w] + d/dw [12sin(w)]

Step-3: Take the constant coefficients outside the derivative notation with the help of the constant function law of the differential notation.

d/dw [3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)] = 3d/dw [w3] + 12d/dw [w4] – 7d/dw [cos(w)] + 5d/dw [w] + 12d/dw [sin(w)]

Step-4: Now differentiate the above expression with the help of power and trigonometry laws.

d/dw [3w3 + 12w4 – 7cos(w) + 5w + 12sin(w)] = 3 [3 w3-1] + 12 [4 w4-1] – 7 [-sin(w)] + 5 [w1-1] + 12 [cos(w)]

= 3 [3 w2] + 12 [4 w3] – 7 [-sin(w)] + 5 [w0] + 12 [cos(w)]

= 3 [3 w2] + 12 [4 w3] – 7 [-sin(w)] + 5 [1] + 12 [cos(w)]

= 9 [w2] + 48 [w3] – 7 [-sin(w)] + 5 [1] + 12 [cos(w)]

= 9w2 + 48w3 + 7sin(w) + 5 + 12cos(w)

## Final words

Now the confusion about solving the complex differential problems can be cleared by following this post. In this post, we have discussed differential calculus and finding it by using an online calculator and manually.