Fundamental of Differential Calculus

Differential calculus is a branch of calculus that studies the concept of a derivative and its applications. Derivative tells us about the rate at which a function changes at any given point. Differential Calculus is crucial to many scientific and engineering areas since it allows for the estimation of instantaneous rates of change and curve slopes. In this article, we will be discussing about Differential Calculus and its fundamentals, which every students should know.

This article helps learners in understanding of differential calculus, its concepts, and its applications. By the end of this article, readers should be able to understand the fundamentals of derivatives and use them to solve real-world issues.

Differential calculus is a branch of mathematics that deals with the study of the rates at which quantities change. It is primarily concerned with the concept of derivative, which represents the rate of change of a function with respect to a variable.

Differential calculus is used to solve problems that involve non-constant rates of change and has applications in various fields such as physics, engineering, economics, and beyond.

Limits

A limit in calculus is a fundamental concept that describes the behaviour of a function as its input approaches a certain value. Limits are used to define the derivative, integrals, and continuity. Mathematically, the limit of a function f(x) as x approaches a value c is expressed as:

[Tex]lim_{x\to c}f(x)=L[/Tex]

This is read as “the limit of f(x) as x approaches c equals L”. The limit tells us the value that f(x) approaches as x gets arbitrarily close to c.

Continuity

A function is said to be continuous at a point if there is no break, jump, or hole in the graph of the function at that point. For a function f(x) to be continuous at a point x = a, the following conditions must be met:

• f(a) exists and has a finite value.
• The limit of f(x) as x approaches an exists (i.e., the left-hand limit and the right-hand limit at [Tex]x=a[/Tex] are equal).
• The limit of f(x) as x approaches a is equal to f(a).

Derivatives:

The definition of a function’s derivative at a given point is:

[Tex]f'(x) = lim_{h→0}\frac{f(x+h) – f(x)}{h}[/Tex]

This formula basically finds the instantaneous rate of change by taking the limit as h approaches zero and applying it to the average rate of change over an interval (from x to x+h).This equation is based on the idea that when h approaches 0, the slope of the tangent line matches the maximum slope of the secant line.

The secant line is a line that passes through the points [Tex](x, f(x))[/Tex] and [Tex](x+h, f(x+h))[/Tex] on the curve. This secant line has the slope [Tex]\frac{f(x+h) – f(x)}{h}[/Tex] . As h decreases, the secant line approaches the tangent line, and therefore the secant line’s slope approaches that of the tangent line.

Differentiation Notation

Differentiation or the derivative of a function can be represented in various ways. Derivative or Differentiation of a function [Tex]f(x) \,or \,y[/Tex] of x with respect to x can be represented by :

• Leibnz’s notation of Derivative : [Tex]\frac{dy}{dx}[/Tex] or [Tex]\frac{d}{dx}(y)[/Tex]
• Lagrange’s notation of Derivative : y’ or f'(x)
• Newton’s notation of Derivative : [Tex]{\dot{y}}[/Tex]
• Euler’s notation of Derivative : Dy or Df(x)

To find the derivative of more complicated functions, we have some rules that make the derivative more simple and easy. Some of them are:

• Product Rule
• Quotient Rules
• Sum Rule
• Power Rule
• Constant Multiple Rule
• Chain Rule

Let’s discuss these rules in detail as follows:

Product Rule of Derivative

Product rule is a formula for calculating the derivative of a function product of two or more functions. The derivative of a product of two functions is defined as the first function multiplied by the derivative of the second function + the second function multiplied by the derivative of the first function.

If we have two functions, u(x) and v(x), and y = u(x)v(x), then the derivative of y with respect to x is given by:

[Tex]\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx} [/Tex]

This rule applies when working with a function that is a combination of two or more simpler functions. It is a necessary technique in calculus for differentiating complicated functions.

Quotient Rule of Derivatives

Quotient Rule is a calculus technique for determining the derivative of a function whose is the ratio of two differentiable functions. If given function f(x) is Quotient of two functions which are differentiable with respect to x i.e.,

[Tex]f(x)=\frac{h(x)}{g(x)}[/Tex]

Its derivative, written as f'(x), is given by:

[Tex]f'(x) = \frac{g(x).h'(x) – h(x). g'(x)}{{g(x)}^2}[/Tex]

This indicates that the derivative of a function ratio (the first function divided by the second function) is equal to the ratio of (the derivative of the first function times the second function minus the derivative of the second function times the first function) to the square of the second function.

Sum Rule of Derivative

The sum rule of derivative is a formula for calculating the derivative of a function sum of two or more functions. If we have two functions, u(x) and v(x), and y = u(x)+v(x), then the derivative of y with respect to x is given by:

y'(x)=u'(x)+v'(x)

Power Rule of Derivative

Power Rule is a calculus technique for determining the derivative of a function whose have algebraic expression of power n. If given function f(x) = xⁿ, where n is real number, then derivative of the function is given by:

[Tex]f^{\prime}(x)=nx^{(n-1)}[/Tex]

Constant Multiple Rule of Derivative

If we have functions, y (x) = c f(x) where c be a constant, then the derivative of y with respect to x is given by:

[Tex]y^{\prime}(x)=c\times f^{\prime}(x)[/Tex]

Chain Rule of Derivative

Chain rule is a formula for calculating the derivative of a composite function. If we have composite functions, y(x)=f(g(x)) where f and g are function of x and differentiable. then the derivative of y with respect to x is given by:

[Tex]y^{\prime}(x)=f^{\prime}(g(x)).g^{\prime}(x)[/Tex]

Differentiation of some common functions is listed in the following table:

Other then above mentioned functions, there are more function which can be diferntiated in differential calculus. Some of these functions are:

• Logarithmic Functions
• Exponential Functions
• Polynomial Functions

Differentiation of Exponential Function

Differentiation of exponential function is depends on base of the function.

• If base is Euler number (i.e. e = 2.71828), them the derivative of e^x is same e^x .
• If base is not Euler number ( say a),, then the derivative of function a^x is given by a^x ln a where ln a is natural log of a i.e. base is e.

Differentiation of Logarithmic Function

Differentiation of Logarithmic function is depends the base of logarithm same as exponential function.

• If base is Euler constant ( e=2.71828), the the derivative is give by [Tex]\frac{d}{dx}(\ln{x})=\frac{1}{x} [/Tex]
• If base is other than Euler number (i.e. base is not e) say(a), then derivative can be written as [Tex]\frac{d}{dx}(\log_{a}x)=\frac{1}{x\ln{a}}[/Tex], where ln(a) is natural log i.e. base e.

Differentiation of Polynomial Function

Differentiation of Polynomial function can be evaluated by using Power rule of Derivative for all the terms.

Polynomial of order n can be written as

P(x)=a_{0}+a_{1}x+a_{2}x^2+\cdots+a_{n}x^n

Derivative of P(x) can be written as:

[Tex]P'(x)=\frac{d}{dx}{a_{0}+a_{1}x+a_{2}x^2+\cdots+a_{n}x^n}=0+1a_{1}+a_{2}2x+\cdots+a_{n}nx^{n-1}[/Tex]

⇒ [Tex]P'(x)=1a_{1}+2a_{2}x+\cdots+na_{n}x^{n-1}[/Tex]

The first principle of differentiation, also known as the first principle of calculus or the difference quotient, is a fundamental concept in calculus used to find the derivative of a function at a given point. It involves taking the limit of the average rate of change of a function over an interval as the interval approaches zero.

Mathematically, the first principle of differentiation is expressed as follows:

Given a function f(x) , the derivative f'(x) at a point x = a is defined by the limit:

[Tex] f'(a) = \lim_{{h \to 0}} \frac{{f(a+h) – f(a)}}{h} [/Tex]

Some other techniques of differentiations are:

• Implicit Differentiation
• Logarithmic Differentiation
• Parametric Differentiation

Implicit Differentiation

Implicit differentiation is used when you have an equation involving both [Tex]x[/Tex] and [Tex]y[/Tex] that does not explicitly solve for y. In such case you differentiate both sides of the equation with respect to ( x ) and then solve for [Tex]\frac{dy}{dx}[/Tex]

For example:

Given [Tex]x^2 + y^2 = 1[/Tex] differentiate both sides with respect to [Tex]x[/Tex]:

[Tex]\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(1)[/Tex]

Applying the chain rule to [Tex]y^2[/Tex] , we get:

[Tex]2x + 2y\frac{dy}{dx} = 0[/Tex]

Solving for [Tex] \frac{dy}{dx}[/Tex] we find:

[Tex]\frac{dy}{dx} = -\frac{x}{y}[/Tex] .

Logarithmic Differentiation

Logarithmic differentiation is useful when dealing with products, quotients or powers of functions that would be cumbersome to differentiate using standard rules. You take the natural logarithm of both sides of an equation and then differentiate.

Example: Consider [Tex]y=x^x[/Tex].

Solution:

To differentiate this function take the natural log of both sides:

[Tex]\ln(y) = \ln(x^x)[/Tex]

Apply the properties of logarithms:

[Tex]\ln(y) = x\ln(x)[/Tex]

Differentiate implicitly with respect to [Tex]x[/Tex]

[Tex]\frac{1}{y}\frac{dy}{dx} = \ln(x) + 1[/Tex]

Solve for ( [Tex] \frac{dy}{dx}[/Tex]):

[Tex]\frac{dy}{dx} = y(\ln(x) + 1) = x^x(\ln(x) + 1)[/Tex]

Parametric Differentiation

When a curve is defined parametrically by two equations [Tex]x=x(t)[/Tex] and [Tex]y=g(t)[/Tex] , one can use parametric differentiation to find derivative [Tex]\frac{dy}{dx}[/Tex].

The derivative [Tex]\frac{dy}{dx}[/Tex] can be calculated by dividing [Tex]\frac{dy}{dt}[/Tex] by [Tex]\frac{dx}{dt}[/Tex] ;

[Tex]\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/Tex]

For example Let [Tex] x = \cos(t) [/Tex] and [Tex]y = \sin(t)[/Tex], then

[Tex]\frac{dx}{dt} = -\sin(t), \quad \frac{dy}{dt} = \cos(t)[/Tex]

So, the derivative [Tex]\frac{dy}{dx}[/Tex] is

[Tex]\frac{dy}{dx} = \frac{\cos(t)}{-\sin(t)} = -\cot(t)[/Tex]

Differential calculus has lots of applications in domains such as physics, engineering, economics and biology. Some of the common use cases are:

• Rate of Change: Determining how quickly a quantity is changing over time, space, or another variable.
• Optimization: Finding maximum or minimum values of functions, which is useful in optimizing processes or resources.
• Curve Sketching: Analyzing the behavior of functions, including identifying critical points, inflection points, and concavity.
• Physics: Calculating velocity, acceleration, and other kinematic properties of moving objects.
• Economics: Modeling marginal cost, marginal revenue, and profit functions in business and economics.
• Biology: Describing population growth and decay, enzyme kinetics, and other biological processes.
• Engineering: Analyzing stress and strain in materials, designing structures, and optimizing systems.
• Finance: Determining interest rates, analyzing investment portfolios, and modeling financial derivatives.
• Computer Graphics: Generating smooth curves and surfaces, such as in 3D modeling and animation.

Example 1: Find the Derivative of f(x)=x with respect to x by using the definition.

Solution :

By the Fundamental definition of Derivative is given by:

[Tex]f'(x)=lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/Tex]

The given function is f(x)=x.

Then,f(x+h)=x+h

So, [Tex]f'(x)=lim_{h\to0}\frac{x+h-x}{h}[/Tex]

⇒ [Tex]f'(x)=lim_{\to0}\frac{h}{h}[/Tex]

⇒ [Tex]f'(x)=lim_{h\to0}(1)=1[/Tex]

Therefore, Derivative of f(x) is equal to 1.

Example 2: Calculate the Derivative of [Tex]F(t)= 3t^2-5t+2[/Tex] with respect to t.

Solution:

To find the Derivative of given function F(t) use the power rule of derivative
Given function is;

[Tex]F(t)=3t^2-5t+2[/Tex]

Differentiate with respect to t :

[Tex]\frac{d}{dt}F(t)=\frac{d}{dt}(3t^2-5t+2)[/Tex]

By distributive property of Derivative:

⇒[Tex]\frac{d}{dt}F(t)=\frac{d}{dt}(3t^2)+\frac{d}{dt}(-5t)+\frac{d}{dt}2[/Tex]

⇒ [Tex]\frac{d}{dt}F(t)=3\frac{d}{dt}t^2-5\frac{d}{dt}t+\frac{d}{dt}2[/Tex]

⇒ [Tex]\frac{d}{dt}F(t)=3(2t)-5(1)+0[/Tex]

⇒ [Tex]\frac{d}{dt}F(t)=6t-5[/Tex]

Problem 3: Differentiate [Tex]g(x) = \sin(x) \cdot \ln(x)[/Tex].

Solution:

Applying the product rule, [Tex]g^{\prime}(x) = \cos(x) \cdot \ln(x) + \frac{\sin(x)}{x}[/Tex]

Problem 4: If [Tex]h(x) = e^{3x} [/Tex], find ( [Tex]h'(x)[/Tex] ).

Solution:

By the exponential rule, [Tex] h'(x) = 3e^{3x} [/Tex].

Problem 5: Determine the derivative of [Tex]p(x) = \frac{1}{1+x^2}[/Tex].

Solution:

Using the quotient rule, [Tex]p'(x) = \frac{-2x}{(1+x^2)^2}[/Tex]

Problem 6: Find the acceleration of a particle if it’s velocity is give by the function sin^2{t} at time t

Solution:

To find the acceleration of the particle we have to differentiate the velocity function with respect to t.

Velocity, [Tex]v(t)=\sin^{2}(t)[/Tex]

Acceleration, [Tex]a(t)=v^{\prime}(t)[/Tex]

Now,to find the v'(t) ,[Tex]v^{\prime}(t)=2sint(cost)[/Tex]

Hence, the acceleration of particle at time t is given by [Tex]\sin(2t).[/Tex]

Problem 7: Find the critical points of [Tex]f(x) = x^3 – 3x + 2[/Tex].

Solution:

First, find the derivative [Tex]f'(x) = 3x^2 – 3 [/Tex]. Setting this equal to zero gives [Tex]x = \pm 1 )[/Tex]. These are the critical points.

Problem 8: What is the slope of the tangent line to the curve y = x² at x = 3 ?

Solution:

The derivative [Tex]y’ = 2x[/Tex], so at x = 3, the slope is [Tex]2 \cdot 3 = 6 [/Tex]

Q1: Differentiate the following functions:

• [Tex]tan^{-1}({2sin{x^{2}}})[/Tex]
• [Tex]\sinh(x)\cot(e^2logx)[/Tex]
• [Tex]\frac{x^2+x-4}{\sqrt{x^2-1}}[/Tex]

Q2: Find the derivative of y=cosecx by using the Fundamental definition of Derivative.

Q3: Differentiate the functions [Tex]g(x)= \frac{1}{{\sqrt{1-x^2}sinx}}[/Tex].

Q4: Calculate the Derivative with respect to u where function is given by f(g(u)) where f(x) =sinx and g(x)= cosx.

Q5: Write the derivative of equation of circle passing through origin.

What is Differential Calculus ?

Differential Calculus is a branch of mathematics that deals with study of rates of change which quantities change . It primarily involves calculating derivatives and using them to solve problems involving non-constant rates of change .

What is a derivative?

A derivative represents the rate of change of a function with respect to variable. It is slope of the tangent line to the curve of the function at any given point .

What are the basic rules of differentiation?

The basic rules of differentiation are power rule , product rule, quotient rule, and chain rule. These rules help us to finding the derivatives of various functions .

What is the significance of the derivative?

Derivatives are significant in finding the rate of change of one quantity relative to another. For example, it can be used to determine the velocity of an object, which is the rate of change of its position with respect to time .

What is implicit differentiation?

Implicit differentiation is used when a function is not explicitly solved for one variable. Instead of solving for y in terms of x, you differentiate both sides of the equation with respect to x and solve for y’ .