Category: Mathematics Questions

Derivative in mathematics is the rate of change of a function with respect to a variable.

Derivatives are very important to the solution of problems in calculus.  To differentiate a function one must know the rules of derivatives.

The derivative of a constant function is always zero.

Let y\,=\,f\,(x)  is any function.

We can explain it as the measure of the rate at which the value of  changes with respect to the change of the variable x. It is read as the derivative of function f  with respect to the variable x. Now the question is how we can differentiate  \,{{x}^{x}}?

So, regarding this question we can say that there are two ways to find the derivative of {{x}^{x}} .

There is an important point to be noticed here that this function is neither a power function of the form x{{\,}^{\hat{\ }}}\,k where k is any constant nor an exponential function of the form b\,\hat{\ }x  where b is a constant, so we can’t use the differentiation formulas for either of these cases directly.

Symbol of derivative

The symbol for denoting the derivative of a function is  f'(x)  or we can also denote it by \frac{{d(y)}}{{dx}}

Differentiating y\,=\,x\,\hat{\ }\,x is simple but tricky.

Method 1

The best way to solve it is to bring the power down and we can bring the power down only by taking logarithm (ln) to the natural base e on both sides.

  Let\,y\,=\,x\,\hat{\ }\,x

1Taking log both sides ,we get0

ln\,y\,=\,ln\,x

Now taking derivative on both sides, we get

\frac{{d(\ln \,y)}}{{dx}}\,=\,\frac{{d\,(x\,\ln \,x)}}{{dx}}

To differentiate the right hand side, we can use the product rule

Product rule says

• First function d/dx second function   + second function d/dx first function
• Also, According to properties of derivatives: derivative of \log \,x is  1/x   and derivative of x is 1.

So, applying this rule above we get

\begin{array}{l}1/y\,\,\frac{{dy}}{{dx}}\,\,=\,x\,\frac{d}{{dx}}\,(ln\,x)\,+\,ln\,x\,\frac{d}{{dx}}\,(x)\\1/y\,\,\frac{{dy}}{{dx}}\,\,=\,\,x(1/x)\,\,+\,\,ln\,x\,(1)\\1/y\,\,\frac{{dy}}{2}\,\,=\,\,1+\,ln\,x\\\frac{{dy}}{{dx}}\,\,=\,y\,(1\,+\,ln\,x)\\\frac{{dy}}{{dx}}\,\,=\,\,x\hat{\ }x\,(1+ln\,x)\,\,\,[as\,the\,value\,of\,y\,=\,x\,\hat{\ }\,x]\end{array}

So , derivative of  {{x}^{x}}\,is\,{{x}^{x}}(1\,+\,ln\,x) by applying logarithm method as this can’t be differentiated with the usual differential methods of derivatives such as product rule,  chain rule, quotient rule etc.

Method 2

There is one more method of finding its derivative by using exponential and logarithm method.

We know that  {{e}^{{ln\,x\,}}}\,=\,x

So, this is same as  {{x}^{x}}\,as\,\,({{e}^{{ln\,x\,}}})\,\hat{\ }x

Which is also same as  e\,\hat{\ }\,(x\,ln\,(x))\,or\,\,{{e}^{{x\,ln\,x\,}}} [using the properties of exponential and logarithmic functions]

We know that,

Hence , by using the formula \frac{d}{{dx}}\,{{e}^{u}}\,=\,{{e}^{u}}\,\frac{{du}}{{dx}}

 

\frac{d}{{dx}}(\,{{e}^{{x\,lnx}}}\,)\,=\,{{e}^{{xlnx}}}\,\,\frac{d}{{dx}}\,(xln\,x)

 

\frac{d}{{dx}}(\,{{e}^{{x\,lnx}}}\,)\,\,=\,{{e}^{{xlnx}}}\,\,(ln\,(x)\,+\,1) [using product rule and also derived in method 1]

Finally rewriting it will give the same answer i.e.

d/\,dx\,\,(\,{{x}^{x}}\,)\,\,=\,(1\,+\,ln\,x\,)\,\,[as\,{{e}^{{x\,ln\,x\,}}}\,is\,nothing\,buy\,\,\,{{x}^{x}}]

 

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To find the sum of factors of 21 first we have to find the factors of 21 .

Factors of 21 are those integers that can divide 21 evenly into equal parts.

Factors of a number include those numbers at which that number is divisible. The concept can be more cleared from the following points:

• The factor of 21 is that number that will divide 21 completely i.e. where the remainder is zero.
• The number 21 is composite. Composite numbers are those numbers that have more than two factors.
• The factors of 21 are all the integers that divide 21 without any remainder.

According to the definition the factors of 21 are 1, 3, 7 and 21 because these are the numbers that divides 21 without any remainder.  Let’s understand how to calculate them.

How to calculate the factors of 21?

By following the below mentioned steps we can find the factors of 21.

Step 1: First , write the number 21

Step 2: Find the two numbers which results in 21  after multiplying them  like 3 and 7 , such that 3 * 7 = 21

Step 3: We know that 3 and 7 are prime numbers with only two factors  that is one and the number itself .

The factors of 3 = 3 * 1

The factors of 7 = 7 * 1

So, we cannot further factorize them.

Step 4: Therefore the factorization of 21 can be expressed as 21 = 3 * 7 * 1

Step 5: Finally, write down all the unique numbers which we can obtain from the above process.

Hence the factors of 21 are 1, 3, 7 and 21.

Pair Factors of 21

To find the pair factors of 21, we need to group them in such as a way that when we multiply the two we get 21.

Positive Pairs

Positive pairs of 21 can be written as  :

( 1 , 21 )  ,  ( 3 , 7 )  ,  ( 7 , 3 )  , ( 21 , 1 )

Negative Pairs

Negative pairs of 21 can be written as :

( – 1 , -21 )  , ( – 3 , – 7 )  , ( -7 , – 3 )  ,   ( – 21 , – 1 )

Out of the factors of 21   numbers 3 and 7 are prime numbers.

Another way of finding factors: Prime Factorization of 21

A prime factorization is the result of factoring a number into a set of components of which every member is a prime number.  This is generally written by showing 21 as a product of its prime factors. For 21, this result would be:

21 = 3 * 7 * 1

Also, 1 and the number itself are the factors of every number.

So, from above discussion we can say that the number 21 has 4  factors : 1 , 3 , 7 , 21

In above factors 1 is the smallest and 21 is the largest factor.

Now, we have all the factors of 21. Let’s find their sum.

Sum of factors of 21

Factors of 21 = 1, 3, 7 and 21

Sum of factors of 21 = 1 + 3 + 7 + 21 =  32

Hence, the sum of factors of 21 is  32

Some interesting insights

• The number of prime factors of 21 is 10 (as only 3 & 7 are prime and 1 is neither prime nor composite).
• The sum of all factors of 21 is 32 .
• The product of all factors of 21 is equal to square of 21 or 21 times of 21.

Product of all factors = 1 x 3 x 7 x 21 = 21 x 21 =  = 441

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Core math is that branch of mathematics where the primary focus is on using and applying math to solve authentic problems and real-life scenarios drawn from work and life with a strong emphasis on problem-solving. In core math focus is shifted from rote learning to concepts based learning and exploring various skills, techniques, and approaches to solve the problems in a way that is best.

What is core math for?

It is suitable for students who do wish to develop their mathematical, quantitative, and statistical skills by studying maths. Core maths is best for students who have cleared GCSE and have achieved grade 4 (Grade C) or above and are not interested in studying AS or A-levels. It’s a level-3 or post-16 qualification which can be taken aside A-level (or AS-level) or other level-3 qualifications. The UCAS tariff points are same as that in AS level qualification.

Objectives

The objectives of teaching core math can be discussed as under:

• Deepen the competence in solution by innovative use of methods and techniques.
• Skill building in mathematical reasoning, thinking, analysing and communication.
• Build confidence in interpreting and analysing authentic real situations.

Importance of Core Math:

Core math means to teach kids innovative and creative ways to problem solving and teaching them different skills that makes learning maths easy, fun-filled and interesting. Core maths understands that every child has its own pace of learning and concept grasping ability. No two students are same.

The modern tutors believe to equip the students with such skills that enable the children to solve the problem with a method best suited to them. This is helpful to those students who need to find the answer in a different way because rote learning is not a one-size fits all approach. It’s all applying common sense and adding practicality to the subject by inculcating scientific approach in students.

Common Core math was basically designed to revolutionize the traditional teaching methods. The main objectives are:

• To skill up the graduating high school students so that they can enter the mainstream with a bang and or excel in higher education.
• Smoothen out the differences between individual state practices and curriculums.

How is core math taught?

In core math students are made to learn through collaboration and problem solving – both vital for future studies and teamwork. Group work and discussion is strongly encouraged while studying as it develops fluency and confidence in applied mathematical skills.

Most Core Maths qualifications also include:

• Interpreting best solutions
• understanding sources of error and bias
• Data analysis and variation in statistics
• risk and probability
• Model growth and decay by using exponential functions
• percentage change
• Graph interpretation
• financial maths
• using standard units
• Fermi estimation
• Normal distribution
• correlation
• Evaluating assumptions while problem solving

Core Maths makes the students feel more confident and equips them with skills that help them to meet the advanced mathematical demands of the market as well as open opportunities for employment and higher education.

 

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Definition

Reciprocal in math is a mathematical number which when multiplied by the given number gives product as 1.

For example: Reciprocal of 23 will be 1/23.

The word reciprocal has been derived from the Latin word “reciprocus” which means returning. In other words, when we take the given number as a multiplier and its reciprocal as multiplicand, the product comes out to be 1. It is also known as inverted number or multiplicative inverse of a number.

How to find a reciprocal of a number?

Reciprocal of a given number is obtained by taking an inverse of it or dividing 1 by that number. We can easily obtain the reciprocal of a natural number, negative number, decimal number as well as fractions.

Reciprocal of a number ‘x’ is written as 1/x or x^-1 .

Reciprocal of a natural number

Let us assume x be a natural number, it’s reciprocal will be written as 1/x.

Steps to do it:

Step1: Turn the number into a fraction by making the numerator as the number and the denominator as 1.

Step2:  Now, interchange numerator and denominator. This means number becomes denominator and 1 gets placed in the numerator.

Problem: Find the reciprocal of 39.

Solution: The reciprocal of 39 is 1/39.

Reciprocal of an integer

If x is an integer where x is not equal to 0, the inverse or reciprocal of x is 1/x. For a negative number say  -y, the reciprocal becomes  -1/y.

Problem: Find the reciprocal of -45.

Solution: The reciprocal of 45 is  -1/45.

Reciprocal of a decimal number

Just like integers, the multiplicative inverse of a decimal number can be obtained by writing 1 over it.

Problem: Find the reciprocal of 9.6.

Solution: The reciprocal of 9.6 is 1/9.6.

Reciprocal of a fraction

A fraction comprises of numerator and denominator. For obtaining reciprocal o a fraction, just interchange the upper and lower part of the fraction. This means if a fraction is m/n, its inverse becomes n/m.

Problem: Find the reciprocal of 13/11.

Solution: The reciprocal of 13/11 is 11/13.

After taking reciprocal of a fraction, the improper fraction becomes a proper fraction and vice versa.

Taking inverse of a mixed fraction

We, all know a mixed fraction comprises of a whole number and a fraction. To take the reciprocal of a mixed number, first, convert the mixed number into an improper fraction and then take its inverse.

Let’s first convert mixed number to an improper fraction. For that, multiply the whole number with the denominator of the fraction and then add the numerator to it. This number becomes the numerator and denominator stays as it is.

Problem: Find the reciprocal of  5\frac{{11}}{{20}}

Solution: Let’s convert 5\frac{{11}}{{20}}   into mixed fraction.

Numerator = (5 x 20) + 11 = 100 + 11 = 111

Denominator = 20

Your fraction becomes \frac{{111}}{{20}}

Now, it’s easy to take reciprocal of a fraction which is \frac{{20}}{{111}} .

Some important points to remember

• Zero is the only number that does not has a reciprocal because any number when multiplied by zero gives 0.
• The product of a reciprocal and the number is always unity.
• Taking reciprocal of a reciprocal number will give back the original number.
• Reciprocal of a number is also termed as turning the number upside down.

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Literally speaking, constant means non-changing. Similarly, in mathematics constant refers to an expression or an object that does not change its value irrespective of its usage.

On the other hand, a variable means whose value can change.

For example, refer to this polynomial

5{{r}^{2}}+4r+9

Here, r is the variable, 5 & 4 and 9 are known as constants. 9 is also known as the constant term in this polynomial. We can also say that coefficients of polynomial where the variable is raised to power zero are constants.

This means that the 9 will remain constant irrespective of the value of r. It has no dependency and has independent existence.

We can also say that, all constants are definable and computable numbers.

Constants are everywhere

The value of a constant is fixed. Apart from numbers, there are many other derived constants that were invented by many renowned mathematicians such as e,\pi

Let’s discuss the same with a few examples:

• In the expression 8h + 6, 8 and 6 both are constants and are also known as coefficients of {{h}^{1}} and {{h}^{0}}  .
• In the expression 8b, 8 is the constant.
• -8xy, here -8 is the constant.

Let’s take some real-life examples:

• The number of months in a year i.e. 12 is a constant.
• We have two eyes and 10 fingers. This number can never be changed.

To clear out the confusion let’s draw differences between Constants and Variables

 

	Variables	Constants
1.	The value of a variable changes depending on its usage.	The value of constant remains fixed. It cannot be altered.
2.	These are represented by letters of alphabets and are usually written in lowercases.	The value of a constant is always a number be it natural, integer fraction, rational or irrational number.
3.	A variable’s value is always unknown.	The value is always known.
4.	For example: x, y, a,b, sin θ etc	For example:  4, 5, 1.6,

 

Notable mathematical constants

Apart from numbers that we know, there are some mathematical constants that are either derived or invented by famous scientists and mathematicians. These objects are called mathematical constants. These are:

• πor pi, This symbol denotes that the ratio of a circle’s circumference to the diameter of the circle always comes out be 3.141592653
• i , or the imaginary part of the complex number (a+ib) is always equal to  or  \sqrt{{-1\,or\,{{i}^{2}}\,=\,1.}}
• The value of (Euler’s number) is approx. 2.718281828…
• Zero
• One (1), the successor of zero
• Pythagoras constant or \sqrt{2} is equal to 1.414.

Let’s practice this with some examples:

Problem 1:  Find the constant terms in 4x+3\,=\,9y-10

For that we need to first simplify the equation:

4x-9y\,=\,-10-3; 4x-9y\,=\,\,-13

The constant term is -13.

4x &  -9y are variable terms.

Problem 2: Harry has 5 apples and Thea has 12 apples more than Benjamin. Find the constants and variables in this scenario?

Let’s understand how many apples each has:

Harry has 5 apples.

Say, Benjamin has  apples

So, Thea has x+12 apples.

The only constant here is 5, as the number of apples owned by Thea and Benjamin depend on variable .

 

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A coefficient is basically defined as an integer which is multiplied with the variable of a single term or the terms of a polynomial. In other words, it refers to a number or quantity placed with a variable. The variables having no number with them are assumed to have 1 as their coefficient.

What is a polynomial?

Poly means ‘many’

Nomial means ‘terms’

A mathematical expression having one or more algebraic terms is called a polynomial. Commonly it is expressed as a sum of several terms having different powers of same variable or variables.

Let us understand with few examples.

In an expression: {ax+bx+z}

Here, is called variable and “a” and “b” are the coefficients and z is called constant.

In an expression: {{x}^{2}}+5 , 1 is the coefficient of {{x}^{2}} .

Now, to identify a coefficient, it should be remembered that it always comes with a variable. In above example of { ax+bx+z, x} is the variable. Hence, ‘a’ and ‘b’ are the coefficient here.

Characteristics of a Coefficient:

A coefficient can be a real/imaginary, positive/negative or even in the form of fractions or decimals.

For example:

In an expression:

• 7x, 4.7 is the coefficient of variable x
• In  \frac{{-1}}{2}y,\,\,\frac{{-1}}{2} is the coefficient of variable y.

Important Points to remember while working on coefficients:

• A coefficient is always attached to a variable, whether variable is a single term or a polynomial.
• In case there is no number or numerical factor in a term, the coefficient of the same is considered a 1.
• The value of the variable is never the same and varies with a situation or according to the question.
• The value of a constant is always fixed and cannot be changed.
• A coefficient can never be zero because when we multiply 0 (as a coefficient) with any variable, the value of the same results into 0.
• While adding or subtracting polynomials, coefficients of the same variables can only participate in these arithmetic operations. But there is no such rule for multiplication and division.

For example :

On adding  8{{x}^{2}}+12y+9   and 17x² – 3y + 4

We get, (8 + 17) x²  + (12 – 3) y + (9 + 4)

The answer becomes 25x² + 9y + 13

Examples:

1. Find out the numerical Coefficients in the term : \frac{{2xy2}}{7}

 

Solution: \frac{2}{7}   is the coefficient of the above term.

2. Find out the numerical coefficients in the following algebraic expression: 7x² – 5y + 8.

 

Solution:  Here we have three algebraic terms. In the term 7x², the numerical coefficient of the term 7x² is 7, -5 the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 7and -5.

 

3. Find out the numerical coefficients in the expression: 9{{a}^{2}}-3y+8

 

Solution: Amongst these three algebraic terms, the numerical coefficient of the term  ~9{{a}^{2}} is 9, -3 is the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 9 and -3.

 

4. Find out the numerical coefficients in the expression: 7{{w}^{2}}+8e+5

 

Solution: In the term 7{{w}^{2}} the numerical coefficient of the term 7{{w}^{2}}   is 7, 8 is the coefficient of e, and 5 is a constant. Therefore, the numerical coefficients are 7 and 8.

 

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