# Calculus

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 1 Introduction 2 Physical Significance of Calculus 3 Functions 4 The Fundamental Theorem of Calculus 5 What are Derivatives? 6 What are Integrals? 7 Examples 8 Summary 9 Frequently Asked Questions (FAQs)

September 5, 2020,

## Introduction

Change is the only constant. Everything around us is always changing. We see things moving, the weather keeps changing, and the population keeps increasing, the temperature of our earth is changing and many such changes.

## Physical Significance of Calculus

Calculus is the branch of Mathematics that can predict how things are going to change. To develop the skill of predicting change we have to understand the basic concepts of Calculus. Differentiation and Integration are the two important ideas around which Calculus is built.
In Calculus, we mostly study things that change according to a pattern. These patterns of change are mathematically represented with the concept of Functions. So the concept of Functions is the main object of study in Calculus. And finally, we will see how to use the ideas of Differentiation and Integration to analyze functions.

## Functions

A function is a rule which maps a number to another unique number. In other words, if we start with an input, and we apply the function, we get an output.
For example, we might have a function that added 3 to any number. So if we apply this function to the number 2, we get the number 5. If we apply this function to the number 8, we get the number 11. If we apply this function to the number x, we get the number x + 3.
We can show this mathematically by writing
f(x) = x + 3
The number x that we use for the input of the function is called the argument of the function. So if we choose an argument of 2, we get f (2) = 2 + 3 = 5. Now let us see how functions are used in Calculus. In case, an area is needed to be found on the graph of a function involving curved lines, Calculus aids in finding the exact area.

## The Fundamental Theorem of Calculus

The main idea in Calculus is called the fundamental theorem of Calculus. Calculus explains about the two different types of calculus:

• Differential Calculus
• Integral Calculus.

Differential Calculus helps to find the rate of change of a quantity whereas integral calculus helps to find the quantity when the rate of change is known.

## What are Derivatives?

Differential Calculus is concerned with the problems of finding the rate of change of a function with respect to other variables. It deals with the variables such as x and y, functions f(x), and the corresponding changes in the variable x and y.
The derivative of a function is represented by or f ‘(x). It means that the function is the derivative of y with respect to the variable x. The symbol dy and dx are called differentials. The process of finding the derivatives is called differentiation.
The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. Now let us find a derivative of a function y=f(x).

To find the derivative of a function y=f(x) we use the slope formula:
Slope = Change in Y × Change in X = Δy × Δx

The slope of a straight line is easy to work out but the slope on the curve is a variable as the line bends. If the curve has to be cut into very small pieces, the curve at the point would look like a very short line.
To work out its slope, a straight line can be drawn through the point with the same slope as the curve at that point. The curve is divided into many points and certain points that were closer to the point of interest approached the actual value of the slope. So the derivative is the slope of a line on a graph.

And (from the diagram) we see that:
x changes from
x to x+Δx

y changes from
f(x) to f(x+Δx)

### Derivative Rules

There are common functions and rules we can follow to find many derivatives. Given below is a set of useful rules to find the derivatives of many functions.

Table of Common Functions and its derivatives

Table of Derivative rules and Functions

### Derivatives of Basic Trigonometric Functions

Trigonometric functions also called circular functions, angle functions or goniometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
The most widely used Trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics.
The basic trigonometric functions include the following six functions:

• sine (sin x)
• cosine (cos x)
• tangent (tan x)
• cotangent (cot x)
• secant (sec x)
• cosecant (csc x)

All these functions are continuous and differentiable in their domains. Below we make a list of derivatives for these functions.

Table of Derivatives of Trigonometric Functions

## What are Integrals?

Integration is the reciprocal of differentiation. It can be said as collecting small parts to form a whole. An integral is definite if it has a specific boundary within which a function needs to be calculated. The lower and upper variable of the independent variable of a function is defined for a definite integral.

### Definite Integral

A Definite integral is denoted as:
ba f(x) dx=F(x)
The symbol for "Integral" is a stylish "S" for "Sum", the idea of summing slices.

### Indefinite Integral

An Indefinite integral does not have a specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:
∫ f(x).dx = F(x) + C

Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. The functions that could have the given function as a derivative are called antiderivatives of the function.
Further, the formula that gives all these antiderivatives is called the indefinite integral of the function and such process of finding antiderivatives is called integration.

The answer to finding the area bounded by a graph of a function would not be right if we could calculate the function at a few points and add up slices of width Δx. To get a better answer we can make Δx a lot smaller and add up many small slices as shown in the figure.

And as the slices approach zero in width, the answer approaches the true answer.
To represent this after the Integral symbol we put the function we want to find the integral of (called the Integrand), and then finish with dx to mean the slices go in the x-direction (and approach zero in width).
And here is how we write the answer:
2x dx= x2 + C
where C is the "Constant of Integration". It is there because of all the functions whose derivative is 2x.

## Examples

 Example 1

Differentiate $$12x^5 + 3x^4 + 7x^3 + x^2 - 9x + 6$$

Solution:

$$\begin{array}{ll}\frac{d y}{d x}=\frac{d}{d x} 12 x^{5}+\frac{d}{d x} 3 x^{4}+\frac{d}{d x} 7 x^{3}+\frac{d}{d x} x^{2}-\frac{d}{d x} 9 x+\frac{d}{d x} 6 & \qquad\text { by sum and difference rule } \\[6pt] \frac{d y}{d x}=12 \frac{d}{d x} x^{5}+3 \frac{d}{d x} x^{4}+7 \frac{d}{d x} x^{3}+\frac{d}{d x} x^{2}-9 \frac{d}{d x} x+\frac{d}{d x} 6 & \qquad\text { by multiple rule } \\[6pt] \frac{d y}{d x}=12\left(5 x^{4}\right)+3\left(4 x^{3}\right)+7\left(3 x^{2}\right)+2 x-9+0 & \qquad\text { by basic derivatives } \\[6pt] \frac{d y}{d x}=60 x^{4}+12 x^{3}+21 x^{2}+2 x-9 & \qquad\text { by simplifying }\end{array}$$

 Example 2

Differentiate $$y = e^{3x − 2}$$

Solution:

\begin{align}\frac{d y}{d x}&=e^{3 x-2} \frac{d}{d x}(3 x-2) \qquad \text{by chain rule}\\[6pt]&=\left(e^{3 x-2}\right)(3) \qquad \qquad\quad\text{by basic derivatives}\\[6pt]&=3 e^{3 x-2}\qquad \quad\;\; \qquad \quad \text{by simplifying}\end{align}

 Example 3

Differentiate $$\text{ln} ( x^2 - 4 )$$

Solution:

\begin{align}\frac{d y}{d x}[f(g(x))]&=f^{\prime}(g(x)) g^{\prime}(x)\\[6pt]f(g(x)&=\ln \left(x^{2}-4\right)=\frac{1}{x^{2}-4} \quad \text { by chain rule }\\[6pt]g(x) & =x^{2}-4=2 x \qquad \qquad\quad \text { by basic derivatives } \\[6pt]\frac{d y}{d x}[f(g(x))] & =\frac{1}{x^{2}-4} \times 2 x \qquad \qquad \; \;\,\text { by simplifying }\\&=\frac{2 x}{x^{2}-4}\end{align}

 Example 4

Differentiate $$y= a^x$$

Solution:

\begin{aligned}&\text { Take natural log on both sides }\\[6pt]&\ln (y)=\ln (x)\\[6pt]&\begin{array}{ll}\operatorname{In}(y)=x(\ln (a)) & \text { by logarithmic rule } \\[6pt]\frac{1}{y}=\frac{d x}{d y}(\ln (a)) & \text { differentiate implicitly }\end{array}\\&\frac{d y}{d x}=y(\ln (a)) \qquad \text { after rearranging }\\[6pt]&\quad\;\;=a^{x} \ln (a) \qquad\quad\;\; \text { since } y=a^{x}\end{aligned}

 Example 5

Differentiate \begin{align}\int_{1}^{2}\left(x^{2}+\frac{1}{x^4}\right) d x \end{align}

Solution:

\begin{align}&=\int_{1}^{2}\left(x^{2}+x^{-4}\right) d x \\[6pt]&=\left[\frac{x^{3}}{3}+\frac{x^{-3}}{-3}\right]_{1}^{2} \quad \text { applying the limits } \\[6pt]&=\left(\frac{8}{3}-\frac{1}{3 \times 8}\right)-\left(\frac{1}{3}-\frac{1}{3}\right) \quad \text { by difference rule }\\[6pt]&=\frac{8}{3}-\frac{1}{24}=\frac{63}{24} \quad \text { by simplifying }\end{align}

 Example 6

Apply implicit differentiation for  $$x^2 + y^2 = 1$$

Solution:

The equation $$x^2 + y^2=1$$ is of the circle with centre as the origin and radius one unit

Differentiate both sides of the equation

\begin{align}2 x+2 y \frac{d y}{d x} &=0 \\x+y \frac{d y}{d x} &=0 \qquad \quad \text { dividing by } 2 \\ y \frac{d y}{d x} &=-x \\\frac{d y}{d x} &=\frac{-x}{y} \qquad \text { Simplifying }\end{align}

This type of differentiation is called implicit differentiation

 Example 7

Evaluate $$\int tan^{-1} x\; dx$$

Solution:

We know that $$\int u dv = uv -\int vdu$$ integration by parts

So take $$u = \text{tan}^{-1}\;x dv = dx$$

$$\therefore \text { du= } \frac{1}{1+x^{2}} \text { and } v=x$$

\begin{align}\therefore \int \tan ^{-1} x d x&=\tan ^{-1} x-\int x \frac{1}{1+x^{2}} d x\\[5pt] &=x \cdot \tan ^{-1} x-\frac{1}{2} \int \frac{2 x d x}{1+x^{2}} \quad \text { by logarithmic rule }\\[5pt] &=x \cdot \tan ^{-1} x - \frac{1}{2} \log \left(1+x^{2}\right)+C \text { where } C \text { is an arbitrary constant }\end{align}

 Example 8

Find the derivative of $$g(x) =\int_{1}^{\sqrt{x}} \frac{s^{2}}{s^{2}+1\text{ ds}}$$

Solution:

\begin{align}&g^{\prime}(x)=\frac{\sqrt{x^{2}}}{\sqrt{x^{2}+1}}\;|\sqrt{x}|\\&g^{\prime}(x)=\frac{x}{x+1} \; \frac{1}{2 \sqrt{x}} \quad \text { applying chain rule }\\&g^{\prime}(x)=\frac{\sqrt{x}}{2(x+1)} \quad \text { by multiplying }\end{align}

## Summary

Using Calculus, we can ask all sorts of questions:

• How does an equation grow and shrink? Accumulate over time?
• When does it reach its highest or lowest point?
• How do we use variables that are constantly changing? (Heat, motion, populations, etc.

And much, much more.
Calculus and Algebra are a problem-solving duo: Calculus finds new equations, and algebra solves them. Like evolution, calculus expands your understanding of how Nature works.

Written by Jesy Margaret, Cuemath Teacher

## What is Calculus used for?

Calculus plays a major role in solving areas of complicated shapes, safety of vehicles, to evaluate survey data for business planning, credit card payment records, or to find how the changing conditions of a system affect us, etc. It is the language of physicians, economists, biologists, architects, medical experts, and statisticians.

## What is the formula for integration by parts?

Integration by parts is a special method of integration that is often useful when two functions are multiplied together. The rule is as follows:
u v dx = uv dx −u' (∫v dx) dx
where u is the function u(x)
v is the function v(x)
u' is the derivative of the function u(x)

## What are the basic rules for integration?

The basic rules of integration include the power, constant-coefficient (or constant multiplier), sum, and difference rules.

## What are the various integration formulae?

The list of integral formulas are

• ∫ 1 dx = x + C
• ∫ a dx = ax+ C
• ∫ xn dx = ((xn+1)/(n+1)) + C ; n≠1
• ∫ sin x dx = – cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec2 dx = tan x + C
• ∫ csc2 dx = -cot x + C
• ∫ sec x (tan x) dx = sec x + C
• ∫ csc x ( cot x) dx = – csc x + C
• ∫ (1/x) dx = ln |x| + C
• ∫ ex dx = e+ C
• ∫ ax dx = (ax/ln a) + C ; a>0,  a≠1

## What are partial derivatives?

A Partial Derivative is a derivative where we hold some variables constant. For a given function, when we find the slope in the x-direction (while keeping y fixed) we will get a partial derivative.

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