Differential Calculus
Differential calculus studies the rate of change of two quantities. Calculus can be divided into two parts, namely, differential calculus and integral calculus. In differential calculus, the derivative equation is used to describe the rate of change of a function whereas in integral calculus the area under a curve is studied.
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.
What is Differential Calculus?
Differential calculus involves finding the derivative of a function by the process of differentiation. The derivative of a function at a particular value will give the rate of change of the function near that value. A derivative is used to measure the slope of a tangent to the graph of a function.
Terms Related to Differential Calculus
Differential calculus is the study of the rate of change of a dependent quantity with respect to a change in an independent quantity. For example, the speed of a moving object can be interpreted as the rate of change of distance with respect to time. If y = f(x) is the function that is differentiated then, according to differential calculus, the notation is given as f'(x) = dy / dx. Some important terms associated with differential calculus are listed below:
Function - A function is defined as a binary relation where each input is mapped to exactly one output. y = 5x + 1 is an example of a function. Here, x (input) is the independent variable, and y (output) is the dependent variable.
Independent variable - In a function, the variable that acts as the input is known as the independent variable. In a mathematical model, the variable that gets manipulated is the independent variable.
Dependent variable - The variable in a function that represents the output is known as the dependent variable. The value of this variable changes with respect to a change in the dependent variable. In other words, the value of a dependent variable is determined by an independent variable.
Domain and Range - In differential calculus, the domain can be defined as the list of all input values while the range is all the output values that are obtained after applying the inputs to a function. For example, y = 5x + 1. Let the domain be {0, 1, 2} then the range will be as follows:
y = 5(0) + 1 = 1
y = 5(1) + 1 = 6
y = 5(2) + 1 = 11
Range = {1, 6, 11}
Limits - A derivative can be defined by the concept of a limit. In differential calculus, a limit describes the value of a function as it approaches a particular input value.
Derivatives - In differential calculus, derivatives are used to find the rate of change of a function. If a tangent line is drawn to a point lying on the graph of a function then the slope of the tangent will give the derivative of the function at the point where the tangent touches the curve. The derivative of a function, f(x), is represented as f'(x), dy / dx, df / dx.
Differential Calculus Example
Suppose there is a function given as f(x) = x2. The slope of this function at a particular point, say 3, can be determined by using differential calculus. The derivative of this function will be f'(x) = 2x. Now x = 3 is substituted in this equation to get f'(x) = 6. Thus, the slope of the tangent line at x = 3 is 6.
Differential Calculus Formulas
The different formulas for differential calculus are used to find the derivatives of different types of functions. According to the definition, the derivative of a function can be determined as follows:
f'(x) = \(lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)
The important differential calculus formulas for various functions are given below:
Elementary Functions
- d/dx sin x = cos x
- d/dx cos x = -sin x
- d/dx tan x = sec2 x , x ≠ (2n + 1) π / 2 , n ∈ I
- d/dx cot x = - cosec2 x, x ≠ nπ, n ∈ I
- d/dx sec x = sec x tan x, x ≠ (2n + 1) π / 2 , n ∈ I
- d/dx cosec x = - cosec x cot x, x ≠ nπ, n ∈ I
- d/dx sinhx = coshx
- d/dx coshx = sin hx
- d/dx tan hx = sec h2x
- d/dx cot hx = -cosec h2x
- d/dx sec hx = -sech hx tan hx
- d/dx cosec hx = -cosec hx cot hx
Inverse Trigonometric Functions
- d/dx sin -1 x = \(\dfrac{1}{\sqrt{(1-x^2}}\), -1 < x< 1
- d/dx cos -1 x = -\(\dfrac{1}{\sqrt{(1-x^2}}\), -1 < x< 1
- d/dx tan -1 x = \(\dfrac{1}{(1+x^2)}\)
- d/dx cot -1 x = -\(\dfrac{1}{(1+x^2)}\)
- d/dx cosec -1 x = -\(\dfrac{1}{|x|\sqrt{x^2 -1}}\), |x| > 1
- d/dx sec-1x = \(\dfrac{1}{|x|\sqrt{x^2 -1}}\)
- d/dx sinh -1 x = \(\dfrac{1}{\sqrt{(x^2+ 1)}}\)
- d/dx cosh -1 x = -\(\dfrac{1}{\sqrt{(x^2-1)}}\)
- d/dx tanh -1 x = \(\dfrac{1}{(1- x^2)}\)
- d/dx coth -1 x = -\(\dfrac{1}{x (1-x^2)}\)
- d/dx cosech -1 x = -\(\dfrac{1}{x\sqrt{1+ x^2}}\)
- d/dx sech-1x = -\(\dfrac{1}{x\sqrt{1- x^2}}\)
Higher Order Derivatives
A derivative is used to give the rate of change of a function. To find the rate of change of this derivative, higher-order derivatives are used. The table given below lists the most commonly used higher-order derivatives for a function, y = f(x), in differential calculus:
| Order of Derivative | First Order Derivative | Second Order Derivative | Third Order Derivative |
|---|---|---|---|
| Notation | f'(x) | f''(x) | f'''(x) |
| Interpretation | \(\frac{\mathrm{d} y}{\mathrm{d} x}\) = \(\frac{\mathrm{d} f(x)}{\mathrm{d} x}\) | \(\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}\) = \(\frac{\mathrm{d}^{2} f(x)}{\mathrm{d} x^{2}}\) | \(\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}\) = \(\frac{\mathrm{d}^{3} f(x)}{\mathrm{d} x^{3}}\) |
| Example: sinx | f'(x) = cosx | f''(x) = - sinx | f'''(x) = - cosx |
Differential Calculus Equations
Differential calculus equations or simply differential equations are equations that relate functions to their derivatives. There are two main types of differential equations, namely, ordinary differential equations and partial differential equations. An ordinary differential equation is one in which there is only one independent variable and the equation contains one or more derivatives with respect to this variable. A partial differential equation consists of one or more independent variables and their partial derivatives.
In differential calculus, there are three general formulas for differential equations. These are given below:
- \(\frac{\mathrm{d} y}{\mathrm{d} x}\) = f(x)
- \(\frac{\mathrm{d} y}{\mathrm{d} x}\) = f(x, y)
- \(x_{1}\frac{\partial y }{\partial x_{1}} + x_{2}\frac{\partial y }{\partial x_{2}} = y\)
Differential Calculus Rules
If the derivatives of certain simple functions are known then the differential calculus rules can be used to find the derivatives of complicated functions. The rules of differential calculus are listed in the table given below.
| Differential Calculus Rule | Form of Function | Interpretation |
|---|---|---|
| Constant Rule | y = c | dy / dx = 0 |
| Constant Multiple Rule | y = cf(x) | dy / dx = cf'(x) |
| Power Rule | y = xn | dy / dx = n · xn-1 |
| Generalized Power Rule | y = [f(x)]n | dy / dx = n[f(x)]n-1f'(x) |
| Sum of Two Functions | y = f(x) + g(x) | dy / dx = f'(x) + g'(x) |
| Difference of Two Functions | y = f(x) - g(x) | dy / dx = f'(x) - g'(x) |
| Product Rule | y = f(x) . g(x) | dy / dx = f'(x)g(x) + f(x)g'(x) |
| Quotient Rule | y = f(x) / g(x) | dy / dx = \(\frac{f'(x)g(x)-g'(x)f(x)}{[g(x)]^{2}}\) |
| Chain Rule for Composite Functions | y = f[g(x)]; y = f(u), u = g(x) | dy / dx = \(\frac{\mathrm{d} y}{\mathrm{d} u}.\frac{\mathrm{d} u}{\mathrm{d} x}\) = f'[g(x)]g'(x) |
Differential Calculus vs Integral Calculus
Differential calculus uses differentiation to find the derivative of a function while integral calculus uses integration to find the integral of a function. Integration is the reverse process of differentiation. The main points of difference between differential calculus and integral calculus are listed in the table given below:
| Differential Calculus | Integral Calculus |
|---|---|
| In differential calculus, derivatives are used to determine the instantaneous rate of change of a function. | Integral calculus uses integrals to determine the area under a curve. Integrals are also known as antiderivatives. |
| Intuitively, differentiation is the process of dividing something into smaller parts to track changes. | Integration sums up infinitesimal pieces to get the total area under a curve. |
| Differential calculus is used to determine if a function is increasing or decreasing. | Integral calculus is used to find areas, volumes, and central points. |
| Example: Differentiate f(x) = x3 f'(x) = 3x2 |
Example: Integrate f(x) = x3 F(x) = \(\frac{x^{4}}{4}+C\) where, C is the constant of integration |
Differential Calculus Applications
There are wide-ranging differential calculus applications. Most quantitative fields use differential calculus such as complex analysis, functional analysis, abstract algebra, and differential geometry.
- In finance, differential calculus is used in portfolio optimization to choose the best stocks.
- In Biology, differential calculus is used to determine how the population of predators and prey evolves over time.
- In mechanics, velocity and acceleration can be derived from the position function using differential calculus.
- Graphic artists use differential calculus to see how a model behaves under conditions that change rapidly.
☛ Related Topics:
- Differentiation and Integration Formulas
- Implicit Differentiation Formula
- UV Differentiation Formula
Important Notes on Differential Calculus:
- Differential calculus involves the use of derivatives to determine the rate of change in a dependent variable with respect to an independent variable.
- Constant rule, difference rule, sum rule, power rule, etc., are the various rules of differential calculus.
- There are two types of differential calculus equations - ordinary and partial differential equations. These equations help to relate functions to their derivatives.
- Integral calculus involves integration which is the reverse process of differentiation.
FAQs on Differential Calculus
What is the Definition of Differential Calculus?
Differential calculus is a branch of calculus involving the study of derivatives that are used to find the instantaneous rate of change of a function using the process of differentiation.
How Hard is Differential Calculus?
Solving problems on differential calculus becomes easy with crystal clear concepts and constant revision. Understanding and learning the formulas is the key to getting a good score in an examination based on differential calculus.
How to Solve Differential Calculus?
The various rules and formulas of differential calculus are used to solve simple and difficult problems. The steps to solve a differential calculus problem are as follows:
- Identify the type of function.
- Apply the required differentiation rule and formula.
- The resultant will be the derivative of the given function.
How to Understand Differential Calculus?
The best way to build a robust understanding of differential calculus is by instilling a deep-seated knowledge of precalculus. The next step is to read the theory of differential calculus. Finally, questions that have varying levels of difficulty must be solved using the formulas and theoretical concepts to master this topic.
Is Differential Calculus the Same as Differential Equations?
Differential equations form a part of differential calculus. Differentials equations can be defined as equations that contain a function with one or more variables as well as the derivatives or partial derivatives with respect to this variable (s).
What is the Difference Between Differential and Integral Calculus?
Integration is the reverse process of differentiation. Differential calculus is concerned with finding the rate of change of a function while integral calculus deals with finding the area under a curve.
What are the Applications of Differential Calculus?
Differential calculus is used in almost every field. It is used to see the rate of growth or decay of functions as well as to find the optimal value (maxima or minima) of mathematical models.