Differential Calculus

Differential calculus deals with the rate of change of one quantity with respect to another. Or you can consider it as a study of rates of change of quantities. Now let us have a look of calculus definition, its types, differential calculus basics, formulas, problems and applications in detail.

Calculus Definition

In mathematics, calculus is a branch that deals with finding the different properties of integrals and derivatives of functions.  It is based on the summation of the infinitesimal differences. Calculus is the study of continuous change of a function or a rate of change of a function. It has two major branches and those two fields are related to each by the fundamental theorem of calculus. The two different branches are:

  • Differential calculus
  • Integral Calculus

In this article, we are going to discuss the differential calculus basics, formulas, and differential calculus examples in detail.

Differential Calculus Basics

In differential calculus basics, we learn about differential equations, derivatives, and applications of derivatives. For any given value, the derivative of the function is defined as the rate of change of functions with respect to the given values. Differentiation is a process where we find the derivative of a function. Let us discuss the important terms involved in the differential calculus basics.

Functions

A function is defined as a relation from a set of inputs to the set of outputs in which each input is exactly associated with one output. The function is represented by “f(x)”.

Dependent Variable

The dependent variable is a variable whose value always depends and determined by using the other variable called an independent variable. The dependent variable is also called the outcome variable. The result is being evaluated from the mathematical expression using an independent variable is called a dependent variable.

Independent Variable

Independent variables are the inputs to the functions that define the quantity which is being manipulated in an experiment. Let us consider an example y= 3x. Here, x is known as independent variables and y is known as the dependent variable as the value of y is completely dependent on the value of x.

Domain and Range

The domain of a function is simply defined as the input values of a function and range is defined as the output value of a function. Take an example, if f(x) = 3x be a function, the domain values or the input values are {1, 2, 3} then the range of a function is given as

f(1) = 3(1) = 3

f(2) = 3(2) = 6

f(3) = 3(3) = 9

Therefore, the range of a function will be {3, 6, 9}

Limits

The limit is an important thing in calculus. Limits are used to define the continuity, integrals, and derivatives in the calculus. The limit of a function is defined as follows:

Let us take the function as “f” which is defined on some open interval that contains some numbers, say “a”, except possibly at “a” itself, then the limit of a function f(x) is written as:

\(\lim_{x\rightarrow a}f(x)= L\), iff  given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f(x) – L| < e

It means that the limit f(x) as “x” approaches “a” is “L”

Interval

An interval is defined as the range of numbers that are present between the two given numbers. intervals can be classified into two types namely:

  • Open Interval – The open interval is defined as the set of all real numbers x such that a < x < b. It is represented as  (a, b)
  • Closed Interval – The closed interval is defined as the set of all real numbers x such that a ≤ x and x ≤ b, or more concisely, a ≤ x ≤ b, and it is represented by  [a, b]

Derivatives

The fundamental tool of differential calculus is derivative. The derivative is used to show the rate of change. It helps to show the amount by which the function is changing for a given point. The derivative is simply called a slope. It measures the steepness of the graph of a function. It defines the ratio of the change in the value of a function to the change in the independent variable. The derivative is expressed by dy/dx.

Differential Calculus

Graphically, we define a derivative as the slope of the tangent, that meets at a point in the curve or which gives derivative at the point where tangent meets the curve. Differentiation has many applications in various fields. Checking the rate of change in temperature of the atmosphere or deriving physics equations based on measurement and units, etc, are the common examples.

In the same way, there are differential calculus problems which have questions related to differentiation and derivatives. Here in this article, we will provide you with formulas to solve all those problems.

Differential Calculus Formulas

How do we study differential calculus? the differentiation is defined as the rate of change of quantities. Therefore, calculus formulas could be derived based on this fact. Here we have provided a detailed explanation of differential calculus which helps users to understand better.

Suppose we have a function f(x), the rate of change of a function with respect to x at a certain point ‘o’ lying in its domain can be written as;

df(x)/dx at point o

Or df/dx at o

So, if y = f(x) is a quantity, then the rate of change of y with respect to x is such that, f(x0) is the derivative of the function f(x). Also, if x and y varies with respect to variable t, then by the chain rule formula, we can write the derivative in the form of differential equations formula;

Differential Calculus Formulas

Differential Calculus Applications

In mathematics, differential calculus is used,

  • To find the rate of change of a quantity with respect to other
  • In case of finding a function is increasing or decreasing functions in a graph
  • To find the maximum and minimum value of a curve
  • To find the approximate value of small change in a quantity

Real-life applications of differential calculus are:

  • Calculation of profit and loss with respect to business using graphs.
  • Calculation of the rate of change of the temperature
  • Calculation of speed or distance covered such as miles per hour, kilometres per hour, etc.
  • To derive many Physics equations.

Differential Calculus Examples

Go through the given differential calculus examples below:

Example 1: f(x) = 3x2-2x+1

Solution: Given, f(x) = 3x2-2x+1

Differentiating both sides, we get,

f’(x) = 3x – 2, where f’(x) is the derivative of f(x).

Example 2: f(x) = x3

Solution: We know,

\(\frac{\mathrm{d} (x^n)}{\mathrm{d} x}\) = n xn-1

Therefore, f’(x) = \(\frac{\mathrm{d} x^3}{\mathrm{d} x}\)

f’(x)= 3 x3-1

f’(x)= 3 x2

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