Differential Calculus: Definition, Formulas, Rules and Examples - Coding

Differential Calculus is a branch of Calculus in mathematics that is used to find rate of change of a quantity with respect to other. It involves calculating derivatives and using them to solve problems involving non constant rates of change. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications.

In this article, we have tried to provide a brief overview of the branch of Differential Calculus including topics such as limits, derivatives, various formulas for derivatives as well as application of derivatives.

Differential-Calculus

Table of Content

What is Differential Calculus?
What is Limit?
Limit Formulas
Continuity, Discontinuity, and Differentiability of a Function
Derivatives
Differentiation Formulas
Implicit Differentiation
Higher Order Derivatives
Error
Approximation
Inflection Point
Tangent and Normal
Increasing and Decreasing Function
Maxima and Minima
Extreme Value Theorem
First Derivative Test
Second Derivative Test
Differential Equation

What is Differential Calculus?

Differential calculus is a branch of calculus that deals with the study of rates of change of functions and the behavior of these functions in response to infinitesimal changes in their independent variables.

In differential calculus, we study derivatives, differentiation techniques (Power, Product, Quotient, Chain rules), implicit differentiation, higher-order derivatives, applications (optimization, related rates, curve sketching), tangent lines, critical points, extrema (max/min values), and many more.

Important Terms in Differential Calculus

Some of the most common terms used in Differential Calculus are:

Independent Variable

In calculus, the independent variable is the variable whose value doesn’t depend on any other variable. For example, in the equation f(x) = 2x², x is the independent variable.

Dependent Variable

In calculus, the dependent variable is a variable whose value depends on another variable, i.e., the independent variable. For example, in the equation y = f(x) = 2x², y is the dependent variable, as its value can only be determined for some chosen value of x.

Function

In calculus, a function is a mathematical rule or relationship that assigns exactly one output value to each input value. This relationship is denoted symbolically as y = f(x), where x is the independent variable and y is the dependent variable.

What is Limit?

For a function y = f(x), then limit x approaches a for function y = f(x) represents the value function approaches when we approach the input value x = a. In simple words, the limit of any function at a given point tells us about its behaviour at and around the point of consideration. It is given as lim _x⇝a f(x). Limit is unique in nature i.e. for x tends to a, there can’t be two values of f(x).

Left Hand and Right Hand Limit

Left-Hand Limit	[Tex]\lim_{x \to a^{-}}f(x)=\lim_{h \to 0^{-}}f(a-h)[/Tex]
Right-Hand Limit	[Tex]\lim_{x \to a^{+}}f(x)=\lim_{h \to 0^{+}}f(a+h)[/Tex]

Existence of Limit

For Existence of [Tex]\bold{\lim_{x \to a}f(x)} [/Tex],

Both [Tex]\lim_{x \to a^{-}}f(x) [/Tex]and [Tex]\lim_{x \to a^{+}}f(x) [/Tex]exists, and
[Tex]\lim_{x \to a^{-}}f(x) = \lim_{x \to a^{+}}f(x)[/Tex]

Properties of Limits

If there are two functions f(x) and g(x) such that their limits [Tex]\lim_{x \to a}f(x) [/Tex]and [Tex]\lim_{x \to a}g(x) [/Tex]exist then following properties are followed

[Tex]\lim _{x \rightarrow a}[f(x) \pm g(x)][/Tex]	[Tex]\lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)[/Tex]
[Tex]\lim _{x \rightarrow a}[k f(x)][/Tex]	[Tex]k \lim _{x \rightarrow a} f(x),[/Tex]
[Tex]\lim _{x \rightarrow a}[f(x) g(x)][/Tex]	[Tex]\lim _{x \rightarrow a} f(x) \lim _{x \rightarrow a} g(x)[/Tex]
[Tex]\lim _{x \rightarrow a} \frac{f(x)}{g(x)}[/Tex]	[Tex]\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}, [/Tex]provided [Tex]\lim _{x \rightarrow a} g(x) \neq 0[/Tex]
[Tex]\lim _{x \rightarrow a}[f(x)]^{g(x)}[/Tex]	[Tex]\left[\lim _{x \rightarrow a} f(x)\right]^{\lim _{x \rightarrow a} g(x)}[/Tex]
[Tex]\lim _{x \rightarrow a}(g \circ f)(x)[/Tex]	[Tex]\lim _{x \rightarrow a} g[f(x)]=g\left[\lim _{x \rightarrow a} f(x)\right][/Tex]
[Tex]\lim _{x \rightarrow a} \log f(x)[/Tex]	[Tex]\log \left[\lim _{x \rightarrow a} f(x)\right], [/Tex]provided [Tex]\lim _{x \rightarrow a} f(x)>0.[/Tex]
[Tex]\lim _{x \rightarrow a} e^{f(x)}[/Tex]	[Tex]e^{\lim _{x \rightarrow a} f(x)}[/Tex]
[Tex]\lim _{x \rightarrow a}\|f(x)\|[/Tex]	[Tex]\left\|\lim _{x \rightarrow a} f(x)\right\|[/Tex]
[Tex]\lim _{x \rightarrow a} f(x)=+\infty \ or -\infty[/Tex]	[Tex]\lim _{x \rightarrow a} \frac{1}{f(x)}=0[/Tex]

Limit Formulas

Some of the common formulas for limits are:

[Tex]\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}[/Tex]	[Tex]n a^{n-1}, n \in Q[/Tex]
[Tex]\lim _{x \rightarrow 0} \frac{(1+x)^n-1}{x}[/Tex]	[Tex]n, n \in Q[/Tex]
[Tex]\lim _{x \rightarrow 0} \frac{\sin x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin x}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{\tan x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan x}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{\sin ^{-1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin ^{-1} x}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{\tan ^{-1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1} x}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{\sin x^{\circ}}{x}[/Tex]	[Tex]\frac{\pi}{180}[/Tex]
[Tex]\lim _{x \rightarrow 0} \cos x[/Tex]	1
[Tex]\lim _{x \rightarrow a} \frac{\sin (x-a)}{x-a}[/Tex]	1
[Tex]\lim _{x \rightarrow a} \frac{\tan (x-a)}{x-a}[/Tex]	1
[Tex]\lim _{x \rightarrow a} \sin ^{-1} x[/Tex]	[Tex]\sin ^{-1} a,\|a\| \leq 1[/Tex]
[Tex]\lim _{x \rightarrow a} \cos ^{-1} x[/Tex]	[Tex]\cos ^{-1} a,\|a\| \leq 1[/Tex]
[Tex]\lim _{x \rightarrow a} \tan ^{-1} x[/Tex]	[Tex]\tan ^{-1} a,-\infty<a<\infty[/Tex]
[Tex]\lim _{x \rightarrow \infty} \frac{\sin x}{x}=\lim _{x \rightarrow \infty} \frac{\cos x}{x}[/Tex]	0
[Tex]\lim _{x \rightarrow \infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{1-\cos x}{x}[/Tex]	0
[Tex]\lim _{x \rightarrow 0} \frac{e^x-1}{x}[/Tex]	1
[Tex]\lim _{x \rightarrow 0} \frac{a^x-1}{x}[/Tex]	[Tex]\log _e a, a>0[/Tex]
[Tex]\lim _{x \rightarrow 0} \frac{\log _e(1+x)}{x}[/Tex]	1
[Tex]\lim _{x \rightarrow e} \log _e x[/Tex]	-1
[Tex]\lim _{x \rightarrow 0} \frac{\log _a(1+x)}{x}[/Tex]	[Tex]\log _a e, a>0, \neq 1[/Tex]

Learn More Limit Formulas.

Evaluation of Limits

Limits can be solved with different methods depending on the type of form it exhibits for x = a.

Determinate Forms
Indeterminate Forms

Determinate Forms

If at x = a, f(x) yields a definite value then the limit is calculated by \lim_{x \to a}f(x)=f(a).

Indeterminate Forms

If at x = a, f(x) yields a value in the form of 0/0, ∞/∞, ∞-∞, 0⁰,1^∞, and ∞⁰ then they are called Indeterminate Forms. It can be solved by following mentioned methods:

Factorization Method

It is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 then x-a is a factor of the numerator and denominator which can be cancelled to make it into determinate form and then solve.

Rationalization Method

This method is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 or ∞/∞ and the denominator is in square root form. In this case, the denominator is rationalized.

Substitution Method

In this case, the x in f(x) is replaced with x = a + h or a – h such that when x tends to a then h tends to 0.

When x→∞: In this case when [Tex]\lim_{x \to ∞}\frac{f(x)}{g(x)} [/Tex]takes the form of ∞/∞ then the numerator and denominator are divided by the highest power of x.

Learn More, Strategy in Finding Limits

L Hospital Rule

L Hospital Rule states that if f(x)/g(x) is in the form of 0/0 or ∞/∞ for x = a then \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)} , where f'(x) and g'(x) are the first order derivatives of functions f(x) and g(x) respectively.

Sandwich Theorem

Sandwich Theorem states that for given functions f(x), g(x), and h(x) that exists in the order f(x) ≤ g(x) ≤ h(x) for x belonging to the common domain then for some value ‘a’ if [Tex] \bold{\lim_{x \to a}f(x)} = p = \bold{\lim_{x \to a}h(x)} [/Tex] then \bold{\lim_{x \to a}g(x)} = p

Continuity, Discontinuity, and Differentiability of a Function

The conditions for continuity, discontinuity, and differentiability of a function at a point are tabulated below:

Continuity	Discontinuity	Differentiability
[Tex]\lim_{x \to a^{-}}f(x)=\lim_{x \to a^{+}}f(x)=f(a)[/Tex]	f(a) not defined Either Left Hand Limit or Right LImit doesn’t exist or infinite [Tex]\lim_{x \to a^{-}}f(x)\neq\lim_{x \to a^{+}}f(x)[/Tex] [Tex]\lim_{x \to a^{-}}f(x)=\lim_{x \to a^{+}}f(x)\neq{f(a)}[/Tex]	[Tex]\lim_{x \to a}\frac{f(x)-f(a)}{x-a} [/Tex]has finite value If Left Hand Derivative Right Hand Derivative i.e. [Tex]\lim_{x \to a^{-}}\frac{f(x)-f(a)}{x-a}=\lim_{x \to a^{+}}\frac{f(x)-f(a)}{x-a}[/Tex]

Fundamental Theorems of Continuity

If f and g are continuous functions then f±g, fg, f/g (where, g≠0), and cf(x) all are continuous.
If g is continuous at ‘a’ and f is continuous at g(a) then fog is also continuous at ‘a’.
If f is continuous in its domain then |f| is also continuous in its domain
If f is continuous in the domain D then 1/f is also continuous in D-{x:f(x) = 0}.

Fundamental Theorems of Differentiability

If f and g are two differential functions then their sum, difference, product, and quotients are also differentiable.
If f and g are two differential functions then fog is also differentiable.
If f and g are not differential functions then their sum and product functions can be differential functions.

Learn More Continuity and Discontinuity

Derivatives

Derivative is defined as the change in the output of a function with respect to the given input. This change is used to analyze the various physical factors associated with the function. Now we will look at the basic expression of Derivatives

Derivative at a Point	[Tex]\bold{\lim_{x \to c}\frac{f(x)-f(c)}{x-c}}[/Tex]
Derivative of a Function	[Tex]\bold{f'(x) = \lim_{h \to 0} \frac{f(x + h) -f(x)}{h}}[/Tex]
Derivative as a Rate Measure	[Tex]\dfrac{dy}{dx}=\bold{\lim_{\Delta x \to 0}\dfrac{\Delta y}{\Delta x}}[/Tex]
Differentiation from the First Principle	[Tex]\bold{f'(x) = \lim_{h \to 0} \frac{f(x + h) -f(x)}{h}}[/Tex]

Algebra of Derivatives

In this section, we will learn how to find the derivative of two functions given in the form of an algebraic expression.

Sum Rule	{f(x) + g(x)}’ = f'(x) + g'(x)
Difference Rule	{f(x) – g(x)}’ = f'(x) – g'(x)
Product Rule	{f(x).g(x)}’ = f'(x).g(x) + f(x).g'(x)
Quotient Rule	{f(x)/g(x)}’ = {f'(x).g(x) – f(x).g'(x)}/(g(x))²

Learn More, Algebra of Derivatives.

Differentiation Formulas

Some of the most common formula used to find derivative are tabulated below:

d/dx(c)	0
d/dx{c.f(x)}	c.f'(x)
d/dx(x)	1
d/dx(xⁿ)	nx^n-1
d/dx{f(g(x))}	f'(g(x)).g'(x)
d/dx(a^x)	a^x.ln(a)
d/dx{ln(x)} {Note: ln(x) = log_e(x)}	1/x, x>0
d/dx(log_ax)	1/xln(a)
d/dx(e^x)	e^x
d/dx{sin(x)}	cos(x)
d/dx{cos(x)}	-sin(x)
d/dx{tan(x)}	sec²x
d/dx{sec(x)}	sec(x).tan(x)
d/dx{cosec(x)}	-cosec(x).cot(x)
d/dx{cot(x)}	-cosec²(x)
d/dx{sin^-1(x)}	1/√(1 – x²)
d/dx{cos^-1(x)}	-1/√(1 – x²)
d/dx{tan^-1(x)}	1/(1+x²)

Solved Examples

Basic Differentiation

Problem: Find dy/dx if y = 3x² – 2x + 5

Solution:

Using the power rule and constant rule:

dy/dx = 3(2x) – 2 + 0

dy/dx = 6x – 2

Product Rule

Problem: Differentiate y = (x² + 1)(x³ – 2)

Solution:

Using the product rule: d/dx[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Let u(x) = x² + 1 and v(x) = x³ – 2

u'(x) = 2x

v'(x) = 3x²

dy/dx = (2x)(x³ – 2) + (x² + 1)(3x²)

dy/dx = 2x⁴ – 4x + 3x⁴ + 3x²

dy/dx = 5x⁴ + 3x² – 4x

Quotient Rule

Problem: Find the derivative of y = (x² + 3) / (x – 1)

Solution:

Using the quotient rule: d/dx[u(x)/v(x)] = [v(x)u'(x) – u(x)v'(x)] / [v(x)]²

Let u(x) = x² + 3 and v(x) = x – 1

u'(x) = 2x

v'(x) = 1

dy/dx = [(x – 1)(2x) – (x² + 3)(1)] / (x – 1)²

dy/dx = (2x² – 2x – x² – 3) / (x – 1)²

dy/dx = (x² – 2x – 3) / (x – 1)²

Chain Rule

Problem: Differentiate y = sin(x²)

Solution:

Using the chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Here, f(x) = sin(x) and g(x) = x²

dy/dx = cos(x²) * 2x

Implicit Differentiation

Problem: Find dy/dx if x² + y² = 25

Solution:

Differentiate both sides with respect to x:

2x + 2y(dy/dx) = 0

Solve for dy/dx:

2y(dy/dx) = -2x

dy/dx = -2x / (2y)

dy/dx = -x / y

Logarithmic Differentiation

Problem: Find the derivative of y = x^(sin x)

Solution:

Take the natural log of both sides:

ln(y) = ln(x^(sin x))

ln(y) = (sin x)(ln x)

Differentiate both sides:

(1/y)(dy/dx) = (cos x)(ln x) + (sin x)(1/x)

Multiply both sides by y:

dy/dx = x^(sin x) [cos x ln x + (sin x)/x]

Related Rates

Problem: A spherical balloon is being inflated at a rate of 2 cm³/s. How fast is the radius increasing when the radius is 5 cm?

Solution:

Volume of a sphere: V = (4/3)πr³

dV/dt = 2 cm³/s (given)

r = 5 cm (given)

Differentiate V with respect to t:

dV/dt = 4πr²(dr/dt)

Substitute the known values:

2 = 4π(5²)(dr/dt)

dr/dt = 2 / (100π) ≈ 0.00637 cm/s

Optimization

Problem: Find the dimensions of a rectangle with perimeter 100 m that has the maximum possible area.

Solution:

Let width = x and length = y

Perimeter equation: 2x + 2y = 100

Solve for y: y = 50 – x

Area function: A = xy = x(50 – x) = 50x – x²

Find dA/dx:

dA/dx = 50 – 2x

Set dA/dx = 0 and solve:

50 – 2x = 0

x = 25

The maximum occurs when x = 25, so y = 25 as well.

Check,

Practice Problems

1).Find dy/dx if y = 4x³ – 2x² + 5x – 7

2).Differentiate y = (x² + 2)(x³ – 3x)

3).Find the derivative of y = (x² – 1) / (x + 2)

4).Differentiate y = sin(2x³)

5).Find dy/dx if x² – xy + y² = 16

6).Differentiate y = e^(x²)

7).Find the derivative of y = ln(x² + 1)

8).A cone-shaped water tank is being filled at a rate of 5 m³/min. If the radius of the base is always twice the height, how fast is the height increasing when the height is 3 meters?

9).Find the absolute maximum and minimum values of f(x) = x³ – 3x² + 1 on the interval [0, 3]

10).Find dy/dx if y = (sin x)^(cos x)

Differentiation of Function of a Function

It says that if f(x) and g(x) are differentiable functions then fog is also differentiable.

d/dx{(fog)(x)} = d/dx{fog(x)}.d/dx{g(x)}

OR

y = g(x) and z = f(y), then dz/dx = (dz/dy).(dy/dx)

Read More about Composition of Function.

Chain Rule

When we need to differentiate the function of a function, we apply the chain rule. In Chain Rule, we first differentiate the first function and then differentiate the second function and write their derivatives in product form. Some of the examples are mentioned below:

d/dx[{f(x)}ⁿ]	n[{f(x)}^n-1].f'(x)
d/dx[ln{f(x)}]	{1/f(x)}.f'(x)
d/dx{e^f(x)}	e^f(x).f'(x)
d/dx{sin(f(x))}	cos(f(x)).f'(x)
d/dx{cos(f(x))}	-sin(f(x)).f'(x)
d/dx{tan(f(x))}	sec²(f(x)).f'(x)
d/dx{tan^-1(f(x))} = {1/(1+(f(x))2)}.f'(x)	{1/(1+(f(x))²)}.f'(x)

Differentiation of a Function with Respect to Another Function

Let’s say we have two functions, u = f(x) and v = g(x) then, differentiation of u with respect to v is found in the following manner

Step 1: Find du/dx

Step 2: Find dv/dx

Step 3: du/dv = (du/dx)/(dv/dx)

Differentiation of Determinant

Let’s say we have to differentiate a determinant given in terms of x, then its derivative is given as differentiation of row (or column) at a time.

If [Tex]\bold{|x| = \begin{vmatrix}f(x)&g(x) \\ u(x)&v(x)\end{vmatrix}}[/Tex]

Then [Tex]\bold{|x|’ = \begin{vmatrix}f'(x)&g'(x) \\ u(x)&v(x)\end{vmatrix} + \begin{vmatrix}f(x)&g(x) \\ u'(x)&v'(x)\end{vmatrix}}[/Tex]

Where |x|’ represents the first order derivative fo |x|.

Implicit Differentiation

Implicit Differentiation is used when a function is not defined explicitly in terms of only one independent variable. In this case, the function is given as g(x,y). It should be noted that here y is equal to f(x). Hence, the differentiation is done in the following manner:

Let’s say we have a function g(x,y) = x² + y + 3xy

then g'(x,y) = d{g(x,y)}/dx = 2x + y’ + 3y + 3xy’

⇒ g'(x,y) = 2x + 3y + y'(1 + 3x).

In implicit differentiation, the chain rule is used, and also product or quotient rule is used wherever applicable. For Example, in the case of 3xy, the product rule is used as x and y are in a product form which gives differentiation of 3xy as 3y + 3xy’.

Higher Order Derivatives

Higher Order Derivatives refer to the derivative of derivative of a function. In this, we first differentiate a function and find its derivatives and then again differentiate the derivative obtained for the first time. If differentiation is done two times then it is called Second Order Derivative and if done for ‘n’ times it is called n^th order derivative. For a function defined as y = f(x), its higher-order derivatives are given as follows:

Second Order Derivative	f”(x) = f²(x) = d²y/dx² = {f'(x)}’
n^th Order Derivative	fⁿ(x) = dⁿy/dxⁿ = {f^(n-1)(x)}’

Error

Differential Error is a method used to calculate error in output for a change in input. To calculate the differential error, follow the below-mentioned steps

Step 1: Take x as the independent variable and x + Δx as the change in the variable.

Step 2: Find Δx by (x + Δx) – x. Assume dx = Δx.

Step 3: Find dy/dx at the given value of x.

Step 4: dy = (dy/dx).dx

The dy obtained in step 4 gives the value of Δy.

Now find y for the given value of x as y = f(x). Thus, the change in y is given as y + Δy.

To find the percentage error multiply (Δy/y) with 100.

Approximation

Approximation is used to find the approximate value of non-perfect square roots or cube roots. To find the approximate value, use the following steps:

Step 1: Define the function. For example, if you have to find the approximate value of a square root take y = f(x) = √x.

Step 2: Take a value closer to the given value of x which is a perfect square or cube.

Step 3: Find the difference between the assumed value and the given value and take it as Δx = dx.

Step 4: Find (dy/dx) at the assumed value

Step 5: Now, dy = (dy/dx).dx

Step 6: dy = Δy is the change in y.

Hence, the approximate value is given as y + Δy.

Critical Point

Critical Point is the point where the derivative of the function is either zero or not defined. C is the critical point of the function f(x) if

(dy/dx)_{x= C} = 0

OR

(dy/dx)_{x= C} = Not Defined

Concave Up and Concave Down

The condition for concave up and down is tabulated below:

Concave Up	f”(x) > 0 at x = a
Concave Down	f”(x) < 0 at x = a

Inflection Point

The point at which the concavity of a function changes is called the Inflection Point. The below-mentioned steps can be used to find the inflection point:

Step 1: Find the second-order derivative of the function i.e. f”(x).

Step 2: Equate f”(x) = 0 and find x.

Step 3: Take a point smaller than x and one larger than x.

Step 4. Put the values in f”(x) and observe the sign. If the sign changes from a value lower than x to a value larger than x then an inflection point exists.

Tangent and Normal

For a curve defined by function f(x) and let us assume there is a Point P(x₁,y₁) on it. Then,

Slope of the Tangent	(dy/dx)_{x = P}
Slope of Normal	-1/Slope of tangent at P = -1/(dy/dx)_{x = P}

Learn More, Tangent and Normal

Increasing and Decreasing Function

Let f(x) is a function differentiable on (a,b) then the function is

Increasing	f'(x) > 0
Decreasing	f'(x) < 0
Constant	f'(x) = 0

Learn More, Increasing and Decreasing Function

Maxima and Minima

The condition for maxima and minima is given below:

Absolute Maximum	x = a	f(x) ≤ f(a) where x and a belong to domain of f.
Absolute Minimum	x = a	f(x) ≥ f(a) where x and a belong to domain of f.
Local Maxima	x = a	f(x) ≤ f(a) for all x near to a.
Local Minima	x = a	f(x) ≥ f(a) for all x near to a.

Learn More, Absolute Maxima and Minima and Relative Maxima and Minima

Fermat’s Theorem

Fermat’s Theorem states that if a function is differentiable at its local extremum then its derivative at that point must be zero i.e., if x = a is the local extrema of f(x) then f'(a) = 0.

Extreme Value Theorem

If f(x) is continuous in the closed interval [a,b] then there exists c ≥ a for which f(c) is minimum and d ≤ b for which f(d) is minimum. In short, for a ≤ c and d ≤ b, f(c) is the absolute minimum in the closed interval [a,b] and f(d) is the absolute maximum in the closed interval [a,b].

First Derivative Test

The condition for first derivative test or local maxima and local minima is given as

Local Minima

Local Maxima

If x = a is point of local minima then

f'(a) = 0 and
f'(x) > 0 for points right to x = a and f'(x) < 0 for points left to x = a

If x = a is the point of local maxima then

f'(a) = 0 and
f'(x) > 0 for points left to x = a and f'(x) < 0 for points right to x = a.

Second Derivative Test

If x = a is a critical point of f(x) such that f'(a) = 0 then if

f”(a) < 0	x = a is the point of local maxima
f”(a) > 0	x = a is the point of local minima
f”(a) = 0	x = a may be a point of local maxima or local minima or none.

Mean Value Theorem

Mean Value Theorem states that if a function f(x) is continuous in the closed interval [a,b] and differentiable in the open interval (a,b) then there exists a point c in (a,b) such that

f'(c) = [f(b) – f(a)]/(b-a)

Learn More, Rolle’s Theorem and Lagrange’s Mean Value Theorem

Differential Equation

Differential Equation refers to an equation that has a dependent variable, an independent variable, and a differential coefficient of the dependent variable with respect to the independent variable.

Order and Degree of Differential Equation

Order of Differential Equation

Degree of Differential Equation

Highest Derivative in the Equation

Example: In (dy/dx)² + 3(d²y/dx²)³ Order is 2.

Exponent raised to Highest Derivative

Example: In (dy/dx)³ + 3(d²y/dx²)² Degree is 3.

Solution of First Order and First Degree Differential Equation

The solution of a first-order and first-degree differential equation can be found by different techniques depending upon the category they belong to.

Equation of the Standard Form

If a differential equation is in the form f[f1(x,y)]d{f1(x,y)} + Φ[f2(x,y)]d{f2(x,y)} + … = 0 then each term can be integrated separately. Some of the standard terms can be replaced by the exact differentials mentioned below:

xdy + ydx	d(xy)
dx + dy	d(x + y)
(xdy – ydx)/x²	d(y/x)
(ydx – xdy)/y²	d(x/y)
(xdy + ydx)/xy	d(log xy)
(ydx – xdy)/xy	d(log x/y)
(xdy – ydx)/xy	d(log y/x)
(dx + dy)/x + y	d(log (x + y))
dx/x² – dy/y²	d(1/y – 1/x)

Equation with Separable Variables

In the variable separable method, the equation is transformed into the form f(x)dx = g(y) and then integrated on both sides resulting in ∫f(x)dx = ∫g(y)dy + C.

Equation Reducible to Variable Separable Form

Let’s say we have a differential equation in the form dy/dx = f(ax + by + c) then

assume ax + by + c = p

Now differentiate both sides which results in a + b(dy/dx) = dp/dx

Hence, dy/dx = 1/b(dp/dx – a).

Now we already have dy/dx = f(ax + by + c) = f(p)

⇒ 1/b(dp/dx – a) = f(p)

⇒ dp/dx = b.f(p) + a

⇒ dp/(b.f(p) + a) = dx

Thus, the equation is separated into two variables p and x which now can be integrated.

Homogeneous Differential Equation

A differential equation of the form dy/dx = f(x,y)/g(x,y) where f and g are functions of the same degree is called Homogeneous Differential Equation. These equations are reduced to a variable separable form by substituting y = vx.

Let’s say dy/dx = f(y/x)/g(y/x) = F (y/x)

Put y = vx

⇒ dy/dx = v + x(dv/dx)

⇒ v + x(dv/dx) = F(v)

⇒ dv/(F(v) – v) = dx/x

The above-mentioned form is the variable separable form that can be integrated to get the result. After integration v must be replaced by y/x.

Linear Differential Equation

A differential equation is said to be linear if the dependent variable and its derivative is of first degree. It can be represented in either of the two forms mentioned below:

dy/dx + Ry = S

dx/dy + Px = Q

For the first form, dy/dx + Ry = S, the integrating factor (IF) is e^∫Rdx

Solution is ye^∫Rdx = ∫Se^∫Rdx + C i.e. y.(IF) = ∫S.(IF) + C

For the second form, dx/dy + Px = Q, the integrating factor (IF) is e^∫Pdx

Solution is ye^∫Pdx = ∫Se^∫Pdx + C i.e. y.(IF) = ∫S.(IF) + C

Also, Read

Summary

Differential calculus is a fundamental branch of mathematics that deals with the study of rates of change and slopes of curves. It provides powerful tools for analyzing how quantities change in relation to each other, which has wide-ranging applications in science, engineering, economics, and many other fields. The core concept of differential calculus is the derivative, which represents the instantaneous rate of change of a function. Key topics include basic differentiation rules (such as the power rule, product rule, and quotient rule), the chain rule for composite functions, implicit differentiation, and applications like related rates and optimization problems. Mastery of these concepts allows for the analysis of function behavior, finding maximum and minimum values, and solving real-world problems involving rates of change. The practice problems provided cover these essential areas, offering a comprehensive review of differential calculus techniques and their applications.

Differential Calculus FAQs

What is Calculus?

Calculus is the mathematical study of continuous change, analogous to how geometry is the study of shape and algebra is the study of operations and their applications.

Who is the Father of Differential Calculus?

Isaac Barrow is generally credited with the early development of the derivative, but both Isaac Newton and Gottfried Wilhelm Leibniz are considered the fathers of differential calculus.

What are Some Important Topics in Differential Calculus?

Limits, Derivatives, Applications of Derivatives and Differential Equations are some important topics in differential calculus.

What is Derivative?

Derivative measures how a function changes as its input changes, representing the slope of the tangent line to the function’s curve at a given point.

What is Critical Point?

A critical point is where the derivative of a function is zero or undefined, crucial for identifying local extrema and points of inflection.

Define Differential Equation.

A differential equation relates a function’s derivatives to the function itself, describing how it changes over time or in relation to other variables.

Reffered: https://www.geeksforgeeks.org

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