Category: Maths Notes PPT

Coplanar vectors are defined as vectors that are lying on the same in a three-dimensional plane. The vectors are parallel to the same plane. It is always easy to find any two random vectors in a plane, which are coplanar. Coplanarity of two lines lies in a three-dimensional space, which is represented in vector form. The coplanarity of three vectors is defined when their scalar product is zero. 

 

All about Coplanar Vectors

Coplanar Vectors is the concept that is taught in classes 11 and 12th respectively. The lesson belongs to a mathematical theory which is quite complex but has been explained in an easy and comprehensible way by the subject experts at that will not only help the students to overcome their doubts but also improve their level of knowledge about the concept with ease.  

The students can easily download the pdf available at ’s website along with other study material for no cost. The resources help the students to clear their concerts, learn new insights, understand the complexities, and much more to promote their overall growth.

For the candidates of class 12 it is discussed here that a vector has direction and a magnitude both however, the scalar has only magnitude. so, the Magnitude of a vector is denoted by |a| or a and it is a non-negative scalar.

 

Equality of Vectors

Two vectors say for example- a and b are said to be equally written as a = b, if they have (i) same length (ii) the same or parallel support, and (iii) the same sense.

 

Types of Vectors

There are the following types of Vectors in the chapter Vectors:

(i) Zero or Null Vector– It is a vector that has initial and terminal points as coincident, and is called zero or a null vector. Also, it is denoted by 0.

(ii) Unit Vector- this is a type of Vector whose magnitude is unity and is called a unit vector. This is denoted by nˆ

(iii) Free Vectors– for these Vectors if the initial point of a vector is not specified, then they are said to be a free vector.

(iv) Negative of a Vector– A vector which has the same magnitude as that of a given vector say ‘a’ and the direction opposite to that of ‘a’ is known as the negative of ‘a a’ and is denoted by —a.

(v) Like and Unlike Vectors– the Vectors are said to be like when they have the same direction and unlike when they have opposite direction.

(vi) Collinear or Parallel Vectors– these Vectors have the same or parallel supports and are called collinear vectors.

(vii) Coinitial Vectors– The Vectors having the same initial point are known as coinitial vectors.

(viii) Coterminous Vectors– these are the Vectors having the same terminal point and are called coterminous vectors.

(ix) Localized Vectors– A vector that is drawn parallel to a given vector through a specified point in space is called a localized vector.

(x) Coplanar Vectors– A system of vectors are said to be coplanar if their supports are parallel to the same plane. However, otherwise, they are known as non-coplanar vectors.

(xi) Reciprocal of a Vector- A vector that has the same direction as that of a given vector but the magnitude that is equal to the reciprocal of the given vector is called the reciprocal of ‘a’.

i.e., if |a| = a, then |a-1| = 1 / a.

 

Coplanarity in Theory

Coplanar lines are a very common topic in three-dimensional geometry. In mathematical theory, the coplanarity of three vectors is called a condition where three lines lying on the same plane are referred to as coplanar.

A plane is a two-dimensional figure going into infinity in the three-dimensional space, while we have used the straight lines as vector equations.

 

Conditions for Coplanar Vectors/ Properties of Coplanar Vectors

• If three vectors are coplanar then their scalar product is zero, and if these vectors are existing in a 3d- space. 

• The three vectors are also coplanar if the vectors are in 3d and are linearly independent.

• If more than two vectors are linearly independent; then all the vectors are coplanar. 

So, the condition for vectors to be coplanar is that their scalar product should be 0, and they should exist on 3d; then these vectors are coplanar. 

The equation system that has the determinant of the coefficient as zero is called a non-trivial solution. The equation system that has a determinant of the coefficient matrix as non zero, but the solutions are x=y=z=0 is called a trivial solution.

 

What are Linearly Independent Vectors?

The vectors, v1,……vn, are linearly independent when the non-trivial combination of the vectors is a zero vector. 

a1v1 + … + an vn = 0 where the coefficients a1= 0 ….. an=0.

 

What are Linearly Dependent Vectors?

The vectors, v1,……vn, are linearly dependent when there is at least one non-trivial combination of these vectors, which is equal to zero vector.

 

How do you Know if Two Vectors are Coplanar?

If the scalar triple product of any three vectors is 0, then they are called coplanar. The vectors are coplanar if any three vectors are linearly dependent, and if among them not more than two vectors are linearly independent.

 

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The numbers in the general form can be expanded using positions. Having said that, wondering how long does it take to count to 1 billion? If you do it manually, count to a Billion might take over 100 years”. However, an online calculator can help you calculate 1 billion counts and easily perform the conversion as well, in a matter of seconds. Just how you would take 10 seconds to count to 100, similarly (put your own value in the billion to million calculator) and find out how many millions make 1 billion.

How to Find 1 Billion How Much Million?

In Mathematical domain, the numbers can be easily described with the help of place value. The place value of the numerical digits is represented in two different ways i.e.  The Indian System and the International System. The place value charts significantly help us to determine the positional values of the number.

The order of the place value always goes from right to left. It means that the place value begins from the unit place {one’s (1’s) place}, then moves to tens (10’s), hundreds (100’s), thousands (1,000’s), ten thousand (10,000’s) and so on. Here, we will discuss the value of 1 billion in rupees, how many millions in a billion and how many crores in 1 billion in terms of the Indian system of place value. Also, we will take into account the place value chart for the Indian System as well as the International System.

1 Billion in Rupees Value

From the place value charts, billions are mainly used in the International System of valuing. The equivalent value of 1 billion in rupees (with respect to Indian System) is given as follows:

1 Billion = 1,000,000,000 Indian Rupees (INR)

We can also write it as:

1 Billion = 10,000 Lakhs (Since, 1 lakh = 1, 00,000)

How Many Crores in 1 Billion

Need to know 1 billion is equal to how much crore? 

1 Billion = 100 crores (Since 1 crore = 1, 00, 00,000)

How Many Millions in a Billion

Thinking hard about how many millions are there in 1 billion? If you’re looking to make a move from million to billion, you’ll require to multiply by 1,000. Simply to say, there are 1,000 million in a billion.

1,000,000 × 1,000 = 1,000,000,000

There are six zeroes (0’s) in a million (or say two groups of three zeroes). While, if we see, there are nine zeroes (0’s) in a billion (or say three groups of three zeroes).

Thus, 1 Billion = 1,000 millions

Reference Chart For How Many Millions Make 1 Billion

Solved Examples 

Example:

What is the value of 6.4 billion in crores?

Solution:

Now, we already know that

1 Billion = 100 crores

Thus, 6.4 Billion = 6.4 x 100 crores

6.4 Billion = 640 crores

therefore, the value of 6.4 billion is 640 crores.

Example

Find out the value of 3 billion in lakhs?

Solution:

It is already known that

1 Billion = 10,000 Lakhs

Thus,

3 billions = 3 x 10,000 lakhs

3 Billions = 30,000 lakhs.

Fun Facts 

• A major difference between the Indian number system and the International number system is the position of comas. 

• Millions are written after thousands under the International numbering method, whereas lakhs are written after thousands when in the Indian system. 

• In countries including India, Pakistan, Nepal, Bhutan, Bangladesh, Sri Lanka and Myanmar the Indian numbering system is most commonly used. 

• The term ‘crores’ is mostly used to represent large sums of money in countries including India, Pakistan, Nepal and Bangladesh.

• The Indian number system is represented through—Thousands, Lakhs, Crores.

• International number system represented through – Million, Billion, Trillion

We know that to find the volume of the cube, we have volume = side3, but to find the side of a cube we have to take the cube root of the volume.

 

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are [4^{3} = 4times 4times times 4 = 64] or [8^{3} = 8 times 8 times 8 = 512] etc.

 

Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbols is [sqrt[3]{}] , it is the “radical” symbol (used for square roots) with a little three to mean cube root.

 

The cube root of 216 is a value which is obtained by multiplying that number  three times. It is expressed in the form of [sqrt[3]{216}] . The meaning of cube root is basically the root of a number which is generated by taking the cube of another number. Hence, if the value of [sqrt[3]{216} = times], then [{times}^{3}] = 216 and we need to find here the value of [times].

 

 

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What is Cube Root?

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself. 

 

For Example, 23 = 8, or the cube root of 8 is 2

 

     33 = 27, or the cube root of 27 is 3

 

   43 = 64, or the cube root of 64 is 4

 

The symbol of the cube root is n3  or [sqrt[3]{n}] 

Thus, the cube root of 8 is represented as  [sqrt[3]{8} = {2}] and that of 27 is represented as  [sqrt[3]{27} = {3}] and so on.

 

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number is the inverse operation of cubing a number. 

 

Example: If the cube of a number 53 = 125

 

Then cube root of [sqrt[3]{125} = {5}]

 

As 216 is a perfect cube, cube root of  [sqrt[3]{126}] can be found in two  ways

 

Prime factorization method and Long Division method.

Calculation of Cube Root of 216

Let, ‘n’ be the value obtained from [sqrt[3]{126}], then as per the definition of cubes, [n times n times n = n^{3} = 216]. Since 216 is a perfect cube, we will use here the prime factorisation method, to get the cube root easily. Here are the following steps for the same.

 

Prime Factorisation Method

Step 1: Find the prime factors of 216

[216 = 2 times 2 times 2 times 3 times 3 times 3]

Step 2: 216 is a perfect cube. Therefore, group the factors of 216 in a pair of three and write in the form of cubes.

[216 = {(2 times 2 times 2)} times {(3 times 3 times 3)}]

[216 = 2^{3} times {3}^{3}]

Using the law of exponent, we get;

[a^{m}b^{m} = (ab)^{m}]

We get,

[216 = 6^{3}]

Step 3: Now, we will apply cube root on both the sides 

[sqrt[3]{216} = sqrt[3]{6^{3}}]

Hence,  [sqrt[3]{216} = 6 ]

 

Solved Examples

Example 1: Find the cube root of 512

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number

[512 = 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2] 

Step 2: Form groups of three similar factors

[512 = 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2]

Step 3: Take out one factor from each group and multiply.

[= 2^{3} times 2^{3} times 2^{3}]

[= 8^{3}]

Therefore, [sqrt[3]{512} = 8 ]

 

Example 2: Find the cube root of 1728

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number 1728 

[= 2 times 2 times 2 times 2 times 2 times 2 times 3 times 3 times 3]

Step 2: Form groups of three similar factors

[= 2 times 2 times 2 times 2 times 2 times 2 times 3 times 3 times 3]

Step 3: Take out one factor from each group and multiply.

= [2^{3}] [times] [2^{3}] [times] [3^{3}]

[= 12^{3}]

Therefore,  [sqrt[3]{1728} = 12]

Quiz

1. Using prime factorization, find the value of   [sqrt[3]{1331} ]

2. Using long division method, find the value of  [sqrt[3]{729} ]

 

Simplify Algebraic Cube Root

To simplify algebraic cubic roots, the cubic radical must meet the following requirements:

• There should be no fractional value under the radical sign.

• Under the cube root symbol, there should be no ideal power factors.

• No exponent value should be bigger than the index value when using the cube root symbol.

• If the fraction appears under the radical, the fraction’s denominator should not include any fractions.

When calculating the cube root of any integer, we will look for the components that appear in the set of three. For instance, the cube of 8 is 2. [2 times 2 times 2] is the factor of 8. Cube roots, unlike square roots, should not be concerned with the negative values under the radical sign. As a result, perfect cubes might have negative values. It is worth noting that perfect squares cannot have a negative value.

Use of Cube and Cube Roots

Several mathematical and physical operations employ cubes and cube roots. It’s frequently used to find the solution to cubic equations. To be more precise, cube roots may be used to calculate the dimensions of a three-dimensional object with a given value. Cubes and cube roots are frequently employed in everyday math computations while studying topics such as exponents. Cube is also used to solve cubic calculations and get the dimensions of a cube given its volume.

Cube Root of a Negative Number

The prime factorization method is the best approach to get the cube root of any integer.

• Perform the prime factorization of the provided integer in the case of negative numbers as well.

• Divide the acquired factors into three groups, each containing the equal number of each component.

• To find the cube root, multiply the components in each group.

• It’s simply that adding three negative numbers yields a negative result. It is indicated by the negative sign in conjunction with the cube root of a negative number.

The cube of a number (x) is the resultant number when it is multiplied three times. As a result, the cube of the number (x) is [ x^{3}] or x-cubed. Let’s use the number 5 as an example. We already know that 555 equals 125. As a result, 125 is known as the 5th cube.

Cubes are the result of multiplying a number by itself three times.

Let’s take an example: 3 × 3 × 3 = 27, so 27 is a cube.

The whole number is multiplied three times, just as the sides for a cube.

Here are the few starting cube numbers:

1 (=1×1×1)

8 (=2×2×2)

27 (=3×3×3)

64 (=4×4×4)

125 (=5×5×5) … etc.

What is a Perfect Cube?

A perfect cube is a number that may be written as the product of three integers that are all the same or equal. For instance, the number 125 is a perfect cube since 53 = 555 = 125. However, 121 is not a perfect cube because there is no integer that produces 121 when multiplied three times. A perfect cube, in other terms, is a number whose cube root is an integer.

What is a Cubed Number?

Suppose we multiply a whole number (not a fraction) by itself, and then again by itself. The result is found to be a cubed number. For example, 3 x 3 x 3 = 27.

A very easy way to write a cube of 3 is [3^{3}]. It means that three is multiplied by itself three times.

The easiest way to do this calculation is to do the multiplication first (3×3) and then to multiply the answer by the same number; 3 x 3 x 3 = 9 x 3 = 27.

The cube is also known as the number which is multiplied by its square.

[ n^{3}] is equal to [ n times n^{3}], which is equal to n × n × n.

The cube function is the known function x ↦ x3 (which is often denoted as y = [ x^{3}]) that connects a number to its cube value. It is an odd function, as

[ -n^{3} = – n^{3}]

Finding the Cube of a Negative Number

Cube for a negative number will always be shown negative, just as the cube of a positive number will always be a positive outcome.

For Example, -125 = -5 x -5 x- -5 = (25 x -5) = -125.

 

Just like the whole numbers (integers), it is very easy to cube a decimal number. You just have to multiply the given decimal number three times.

Finding the Cube of a Decimal

How to find the cube root of a number?

The prime factorization method can be used to find a number’s cube root. Begin with determining the prime factorization of the given number to determine the cube root. Then, divide the resulting components into groups of three identical elements. Then, to find the result, delete the cube root symbol and multiply the factors. If any factor remains that cannot be divided equally into groups of three, the provided number is not a perfect cube, and the cube root of that integer cannot be found. The resultant number (product) of multiplying a number three times by itself is known as the cube of the original number. It’s called a cube since it’s used to indicate a cube’s volume. To find the cube of a number, multiply it by itself first, then multiply the result by the original number once more.

Cube of a fraction 

The cube of a fraction can be calculated by multiplying it three times, just as the cube of an integer.

Cube of a negative number 

Finding the cube of a negative number follows the same steps as finding the cube of a whole number or fraction. Always remember that a negative number’s cube is always negative, while a positive number’s cube is always positive.

Properties of Cube root

There is one real number y such that [ y^{3} = x] for any real number x. Because the cube function is rising, it does not produce the same output for two different inputs, and it is applicable to all real values. In other words, it’s a one-to-one bijection. After that, we may define a one-to-one inverse function. We can establish a unique cube root of all real numbers for real numbers. The cube root of a negative number is a negative number if this definition is used.

How to find the cube root of a number?

The inverse technique of determining a number’s cube is called cube root. The symbol ∛ is used to represent it.

List of cube roots 

Using the Division Method to Find the Cube Root

Using the division approach to obtain the cube root is akin to using the long division method or the manual square method. From the back to the front, make a set of three-digit numbers. The next step is to determine which number has a cube root that is less than or equal to the supplied integer. Subtract the acquired number from the specified number, then enter the result in the second box. Following this, get the multiplication factor for the next step in the long division procedure, which is done by multiplying the initial number obtained. To find the cube root of a number, repeat the technique described above. When the given integer is not a perfect cube number, this lengthy division method is utilized. It takes a long time to find the cube root of an integer using this method.

Solved Questions 

1. Calculate the Cube 5 and 6.

Solution: To find the cube of 5, we will multiply 5 three times, i.e., 5× 5 ×5 = 125.

To find the cube of 6, we multiply 6 three times, i.e., 6× 6 ×6 = 216. 

2. Find the Cube of (0.06)³.

Solution: 

0.06 can be written as (6/100)³.

Simplifying (6/100)³, we get 

(3/50)³ = (3)³/(50)³

(3 * 3*3)/(50 * 50* 50)

= 27/125000

Hence, the cube of (0.06) is 27/125000.

What is a Data Set?

Dataset meaning corresponds to the collection of data that is similar. A data set is generally represented in the form of a table. In this table, each column corresponds to a variable and each row corresponds to the values of the variable whose data is collected. Datasets find a huge range of applications in the comparison, prediction, or computation involved in several domains of Statistics, Economics, Mathematics, and Science. The datasets consist of past, present, and future data which can be used for comparisons.

Data Set Meaning:

Handling the large data in various domains of the real-world is a tedious task. It needs the utmost care in the collection, classification, arrangement, and storage of data. Each datum should be classified carefully so that it can be easily fetched when the access is required. The data is therefore arranged in an organized manner in the form of tables or schematic symbols or any other Mathematical objects so that they can be easily accessed when required. This kind of data arranged in a specified pattern is called the data set. 

To understand the data set meaning, let us consider an example of a school. The marks of a student in Class 9 in one of the unit tests conducted in the month of July 1999 is to be rechecked in a school consisting of 3500 students in a year. In this case, the data of marks obtained by the students is classified under different types of data sets like year-wise, class-wise, month-wise, subject wise, etc. This organized arrangement of the data of student’s marks helps the management of the school to fetch the marks easily in a short time. However, if the data is haphazard fetching of the data when required becomes highly complicated.

Sample Data Sets:

Sample data sets are the data sets obtained from a specified statistical population. The process involved in collecting the sample data sets is called the sampling. Any scientific study or research is performed over a set of sample data before it is generalized. For example research is made that a particular medicine is newly invented which is capable of curing cancer. Before this medicine is set out in the market, the medicine is first tested on a few sample data sets such as a sample of rats, a sample of higher animals, and finally the sample data sets of human beings. If the research succeeds with a positive result in all the sample data sets, then it is generalized and released for public use.

Types of Data Sets:

The different types of data sets are:

Numerical data set example: Height and weight of a people, the population of a country, a score of students, etc

Example: Age and systolic blood pressure of a group of people

• Multivariate datasets: The data set consists of data values of more than two variables.

• Categorical datasets: The datasets which represent the specific category of data based on their characteristics are called categorical datasets.

Categorical data set Example: Gender of an animal, state of matter of a substance, etc.

Example: The height and weight of a group of people are interdependent with each other. 

Measures of Central Tendency of Data sets:

Central tendency is a measure of the central value of the collected data in a dataset. There are three measures of central tendency. They are:

1. Mean: It is the average value of all the samples of the given dataset.

2. Median: It is the middle value or midpoint of the values in the sample data sets.

3. Mode: It is the most repeated value in the given sample datasets. 

Data Set Example Problem:

1. The sample values of data in a data set example are given in the table below. Find the mean, median and mode of the given values in the data set.

[7, 9, 4, 7, 6, 4, 3, 1, 7, 7]

Solution:

Values of dataset arranged in ascending order = [1, 3, 4, 4, 6, 7, 7, 7, 7, 9]

Mean of the given dataset is its average

Mean = [frac{Sum of Values}{No. of Values}] = [frac{7+9+4+7+6+4+3+1+7+7}{10}] = [frac{55}{10}] = 5.5

Median is the middle value of the data set. Since the data set consists of 10 values,

Median = [frac{6+7}{2}] = [frac{13}{2}] = 6.5

Mode of a given dataset is the most repeated value in the dataset.

In the given dataset, the value ‘7’ is repeated the maximum number of times.

Mode = 7

Fun Facts about What is Data Set:

• If a data set does not contain any missing values or any aberrate data and can be easily altered, then such a dataset can be regarded as a good dataset.

• Datasets form the basic building blocks of various domains of data mining and data science.

• Central tendency measurements are applicable only for numerical datasets.

• Range of a dataset is the difference between the highest and lowest values in the dataset.

Derivatives are important concepts of Mathematics. Derivatives are basic to the different solutions to the problems of calculus and differential equations. Generally, scientists observe a dynamic system to get the rate of change of some variable of interest, including this information into some differential equation and use integration methods to obtain functions that can be used to estimate the behaviour of the original system in different conditions. Let us now discuss what is derivative in Mathematics?

Derivatives in Mathematics is the rate of change of a function in terms of a variable. The rate of change of a function in derivatives can be estimated by calculating the ratio of the change of the function $Delta b$ to the change of the independent variable $a$. This ratio in the derivative is considered in the limit as $Delta a to 0$.

As you have learned what is derivative in Mathematics, let us now discuss how to define derivatives and derivative rules that can be used to calculate many derivatives.

Define Derivatives

Let $fx$ be a function whose domain includes an open interval at some point $x_0$. Then the function $fa$ is considered to be differentiable at $x_0$, and the derivative of $fx$ at $x_0$ is expressed as:

$f’ x_0 = underset{Delta to 0}{lim}{dfrac{Delta y}{Delta x}}$

$Rightarrow underset{Delta to 0}{lim}{dfrac{f(x_0 + Delta x – f(x_0)}{Delta x}}$

The derivative of a function $y$ in Lagarangee’s form is expressed as:

$y = f(x) text{ as } f ‘(x) text{ or } y’ (x)$

The derivative of a function y in Leibniz’s form is expressed as:

$y = f (x) text{ as } dfrac{df}{dx} text{ or } dfrac{dy}{dx}$

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Derivative Rules

Below are some of the derivative rules that can be used to calculate differentiation questions.

The Constant Rule

Let $y$ be an arbitrary real number. The constant rule is defined as:

$dfrac{d(y)}{dx} = 0$

The Constant Function Rule

Let $y$ be an arbitrary real number, and $g(x)$ be an arbitrary differentiable function. The constant function rule states that

$dfrac{d(y cdot g(x))}{dx} = y cdot g’(x)$

The Power Rule

Let $a$ and $b$ be a real number, with $a neq 0$ and $a$ and $b$. The Power rule states that

[frac{d}{dx} x^{n} = n x^{n-1}]

The Product Rule

Let $a(x)$ and $b(x)$ be an arbitrary differentiable function. The product rule states that

$dfrac{d(a(x) cdot b(x))}{dx} = a’(x) cdot b(x) + a(x) + b’(x)$

The Chain Rule

The derivative of the function $h(x)= a(b(x))$  in terms of chain rule is expressed as:

$h'(x)= a'(b(x)) cdot  b'(x)$.

The product rule in Leibniz’s notation is represented as

$dfrac{dh(x)}{dx} = dfrac{da(b(x))}{db(x)} cdot dfrac{db(x)}{dx}$

The $x$ Rule

Let us consider $x$ as an arbitrary variable, then $X$ rule states that

$dfrac{d(x)}{dx} = 1$

The Sum and Difference Rule

Let $a(x)$ and $b(x)$ be an arbitrary differentiable function. 

Recall that for an arbitrary function $k(x)$,

$dfrac{d(k(x)}{dx} = k’(x) = underset{Delta h to 0}{lim} dfrac{k(x+h)-k(x)}{h}$

The sum rule states that:

$dfrac{d(a(x)+b(x))}{dx} = a’(x)+b’(x)$

The difference rule states that:

$dfrac{d(a(x)-b(x))}{dx} = a’(x) – b’(x)$

The Quotient Rule

Let $a(x)$ and $b(x)$ be an arbitrary differentiable function with $a(x) neq 0$ and $b(x) neq 0$. The quotient rule states that:

$dfrac{dleft({dfrac{a(x)}{b(x)}}right)}{dx} = dfrac{a’(x) cdot b(x) – a(x) cdot b’(x)}{(b(x))^2}$

Fun Facts

• Gottfried Wilhelm Leibniz introduced the symbols $dx, dy$ and $dfrac{dy}{dx}$ in 1675. The symbols are commonly used when the equation $y = f(x)$ is examined as a functional relation between dependent and independent variables.

• The first derivative is represented by $dfrac{dy}{dx}, dfrac{df}{dx}$ or $dfrac{d}{dx} f$, and was once considered as an infinitesimal quotient.

Solved Examples

1. Evaluate $dfrac{d}{dx}( 2x + 1)^2$ using the chain rule.

Solution:

Let $g(x)= (2x + 1)$, and $f(x)= x^2$

Then, $f(g(x))=  (2x + 1)^2$

As we know $f ‘(x)= 2x$, and $g'(x)= 2$.

Accordingly, $dfrac{d}{dx}( 2x + 1)^2 = f'(g(x)) cdot g'(x)$

$= f'( 2x + 1) cdot 2$

$= 2 ( 2x + 1) cdot 2$

$= 8x + 4$

2. Differentiate the function $f(x) = x^{10}$ using power rule.

Solution:

$f'(x)= 10x^{10-1}$ 

$= 10x^9$

3. Find the derivative of the following function:

$y = dfrac{1-y}{y^2 + 2}$

Solution:

We have

$y’ = dfrac{( 1 -k)’ ( k^2 + 2) – ( 1 – k) (k^2 + 2)’}{(k^2 + 2)^2}$

$y’ =  dfrac{(-1) cdot (k^2 + 2)^2 – ( 1 – k)(k^2 + 2)^{2’}}{(k^2 + 2)^2}$

$y’= k^2 – 2k – 2(k^2 + 2)^2$