Definition
Calculus
Calculus, a branch of Mathematics, developed by Newton and Leibniz, deals with the study of the rate of change. Calculus Math is generally used in Mathematical models to obtain optimal solutions. It helps us to understand the changes between the values which are related by a function. Calculus Math mainly focused on some important topics such as differentiation, integration, limits, functions, and so on.
Calculus Mathematics is broadly classified into two different such:
- Differential Calculus
- Integral Calculus
Both the differential and integral calculus deals with the impact on the function of a slight change in the independent variable as it leads to zero. 1
Integral Calculus
Integral Calculus is the study of integrals and their properties. It is mostly useful for the following two purposes: 1
- To calculate f from f’ (i.e., from its derivative). If a function f is differentiable in the interval of consideration, then f’ is defined in that interval.
- To calculate the area under a curve.
The integration formulas have been broadly presented as the following six sets of formulas. Basically, integration is a way of uniting the part to find a whole. The formulas include basic integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and some advanced set of integration formulas. Integration is the inverse operation of differentiation. Thus the basic integration formula is ∫∫ f'(x).dx = f(x) + C
Using the fundamental theorems of integrals, there are generalized results obtained which are remembered as integration formulas in indefinite integration. 2
- ∫ xn dx = x(n + 1)/(n + 1)+ C
- ∫ 1 dx = x + C
- ∫ ex dx = ex + C
- ∫1/x dx = log|x| + C
- ∫ ax dx = ax /log(a)+ C
- ∫ ex[f(x) + f'(x)] dx = ex f(x) + C
Derivative
The word derivative is probably the most common word you’ll be hearing when taking your first differential calculus. This entire concept focuses on the rate of change happening within a function, and from this, an entire branch of mathematics has been established.
Derivative in calculus refers to the slope of a line that is tangent to a specific function’s curve. It also represents the limit of the difference quotient’s expression as the input approaches zero.
Derivatives are essential in mathematics since we always observe changes in systems.
The derivative of a function, represented by dy / dx or f'(x), represents the limit of the secant’s slope as h approaches zero.
In precalculus classes, we’ve learned about secant lines and how we can calculate the rate of change between (x, f(x)) and (x, f(x + h)) using the formula for slopes. 3
Visualizing Derivatives in Calculus
Example
f(x) = x2 − 5x + 6
f'(x) = 2x − 5
Differentiation
Differentiation is a method of finding the derivative of a function. Differentiation is a process where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate change of displacement with respect to time, called velocity. The opposite of finding a derivative is anti-differentiation.
If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx. This is the general expression of derivative of a function and is represented as f'(x) = dy/dx, where y = f(x) is any function. 4
Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.
Rules of Differentiation
If f is differentiable at a point x = x0, then f is continuous at x0. A function is differentiable in an interval [a,b] if it is differentiable at every point [a,b]. The sum, difference, product, and composite of differentiable functions, wherever they are defined, are differentiable, and the quotient of two differentiable functions is differentiable, wherever it is defined. The differentiation rules are listed as follows: 5
- Sum Rule: If y = u(x) ± v(x), then dy/dx = du/dx ± dv/dx.
- Product Rule: If y = u(x) × v(x), then dy/dx = u dv/dx + v du/dx
- Quotient Rule: If y = u(x) ÷ v(x), then dy/dx = (v du/dx- u dv/dx)/ v2
- Chain Rule: Let y = f(u) be a function of u and if u=g(x) so that y = f(g(x), then d/dx(f(g(x))= f'(g(x))g'(x)
- Constant Rule: y = k f(x), k ≠ 0, then d/dx(k(f(x)) = k d/dx f(x).
Example
Differentiate f(x) = 6x3 – 9x + 4 with respect to x.
On differentiating both the sides w.r.t x, we get;
f'(x) = (3)(6)x2 – 9
f'(x) = 18x2 – 9
Differential Calculus
Differential Calculus is concerned with the problems of finding the rate of change of a function with respect to the other variables. To get the optimal solution, derivatives are used to find the maxima and minima values of a function. Differential calculus arises from the study of the limit of a quotient. It deals with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The symbol dy and dx are called differentials. The process of finding the derivatives is called differentiation. The derivative of a function is represented by dy/dx or f’ (x). It means that the function is the derivative of y with respect to the variable x. Let us discuss some of the important topics covered in the basic differential calculus. 1
One of the main uses of differential calculus is in finding the minimum or maximum value of a given function as part of an optimization problem. In this article, we will learn more about differential calculus, the important formulas, and various associated examples. 6
Differential Equation
In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. 7
Example
Verify that the function y = e-3x is a solution to the differential equation
The function given is y = e-3x. We differentiate both the sides of the equation with respect to x,
Now we again differentiate the above equation with respect to x,
We substitute the values of
in the differential equation given in the question,
On left hand side we get, LHS = 9e-3x + (-3e-3x) – 6e-3x
= 9e-3x – 9e-3x = 0 (which is equal to RHS)
Therefore, the given function is a solution to the given differential equation.
Who
“It was not possible to separate in the Martian tongue the human concepts: ‘religion,’ ‘philosophy,’ and ‘science’—and, since Mike thought in Martian,
Robert A. Heinlein, Stranger in a Strange Land
it was not possible for him to tell them apart.”
To misquote Mr. Heinlein: “It was not possible to separate the concepts: ‘integral calculus,’ ‘derivatives,’ ‘differentiation,’ ‘differential calculus,’ and ‘differential equations’—and, since Mike thought he understood English, it was not possible for him to tell them apart.”
What
To make these concepts easier to understand. (They are now for me.)
Why
I have never had a teacher explain these concepts. They were just presented without explanation of how they relate.
See Theoretical Knowledge Vs Practical Application.
How
The References and Additional Reading websites will assist you with understanding the difference between Integral Calculus, Derivatives, Differentiation, Differential Calculus and Differential Equations.
As some professors say: “It is intuitively obvious to even the most casual observer.“
References
1 “Calculus (Differential And Integral Calculus With Examples)”. 2022. BYJUS. https://byjus.com/maths/calculus/.
2 “Integration Formula – Examples | List Of Integration Formulas”. 2022. Cuemath. https://www.cuemath.com/calculus/integration-formulas/.
3 “Derivative Calculus – Definition, Formula, And Examples”. 2022. Story of Mathematics. https://www.storyofmathematics.com/derivative-calculus/.
4 “Differentiation In Calculus (Derivative Rules, Formulas, Solved Examples)”. 2022. BYJUS. https://byjus.com/maths/differentiation/.
5 “Differentiation – Formula, Examples | Differentiation Meaning”. 2022. Cuemath. https://www.cuemath.com/calculus/differentiation/.
6 “Differential Calculus – Definition, Formulas, Rules, Examples”. 2022. Cuemath. https://www.cuemath.com/calculus/differential-calculus/.
7 “Differential Equations (Definition, Types, Order, Degree, Examples)”. 2022. BYJUS. https://byjus.com/maths/differential-equation/.
Additional Reading
“Differential Calculus (Formulas And Examples)”. 2022. BYJUS. https://byjus.com/maths/differential-calculus/.
Sangha. “Where Are Both Differentiation And Differential Equations The Same?” 2022. Quora. https://qr.ae/pvH82W.
Mathematical Mysteries 