Table of Content

• Key Concepts in Differential Calculus
• Limits
• Continuity
• Derivatives:
• Differentiation Notation
• Basic Rules of Differentiation
• Product Rule of Derivative
• Quotient Rule of Derivatives
• Sum Rule of Derivative
• Power Rule of Derivative
• Constant Multiple Rule of Derivative
• Chain Rule of Derivative
• Differentiation of Common Functions
• First Principle of Differentiation
• Techniques of Differentiation
• Applications of Differential Calculus

Function	Derivative (f'(x))
Constant (c)	0
sin x (trigonometric)	cos x
cos x (trigonometric)	– sin x
tan x (trigonometric)	sec2 x
sin-1 x (inverse trigonometric)	1/√(1-x2)
cos-1 x (inverse trigonometric)	-1/√(1-x2)
tan-1 x (inverse trigonometric)	1/(1+x2)

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]

First Principle of Differentiation

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]

Techniques of Differentiation

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]

Applications of Differential Calculus

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.

Solved Examples

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]

Practice Questions

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.

FAQs on Differential Calculus

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’ .