Derivative Step by Step

erivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.-  Source : Wikipedia

Calculus is used to solve differentiation of any subjective equation and the equivalent output result. we need to find f(x) where,     d/dx f(x) = g(x).The derivative of constant is zero in calculus. The linear equations are also compared using calculus. Integration is considered as one of most important study of calculus in mathematics. British mathematician, Isaac Newton and the German man Gottfried Leibnitz, invented the calculus. The methods that are applied in continous graphs, curves or fucntions  is calculus.

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Steps to solve the Derivative function:

Derivative  problem 1:

Find the first derivative and second derivative of

f(x) = x4 + 7x3 - 3x2 - 9x +3

Solution:

Step 1:  First differentiate f(x) = x4 + 7x3 - 3x2 - 9x +3

Step 2:  when we differentiate the first term we get  (1 × 4) x (4-1)

Step 3:  The above equation can above simplified as  4 x 3

Step 4:  Like wise we can differentiate  and simplify all the terms in the equation

Step 5:  Finally after the first derivative we will get  4 x 3 + 21 x2 - 6x - 9

Step 6:  Now we will go for Second derivative
End of the second derivative we will get   12x2 + 42 x - 6


First derivative:

f' = (df )/ (dx)

= (1 × 4) x (4-1) + (7 × 3)x(3-1) - (3 × 2)x(2-1) - (9 × 1)x (1-1)

= 4 x 3 + 21 x2 – 6x - 9

Second derivative:

f '' = (df ' )/ (dx)

= 4 x 3 + 21 x2 – 6x - 9

= 12x2 + 42 x - 6

Answer:

(d^2)/(dx^2)  (x4 + 7x3 - 3x2 - 9x +3) = 12x2 + 42 x - 6

Derivative problem 2:

Find the differential for y = sqrt(2x^3 - 9x)

Solution:

Step 1:  First turn sqrt(2x^3 - 9x)to (2x3 - 9x)1/2
Step 2:  Differentiate in the usual method
Step 3:  Then finally we get,

d/(dx) ((2x3 - 9x)1/2 ) = ½ ( 2x3 - 9x ) ½ -1( 6x2 - 9 )

= ½ ( 2x3 - 9x ) -1/2( 6x2 - 9 )

= ( 6x^2 - 9 ) / (2 sqrt (2x^3 - 9x ))

Answer of given calculus test exam problem is

d/(dx) ((2x3 - 9x)1/2 ) = ( 6x^2 - 9 ) / (2 sqrt (2x^3 - 9x ))

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Derivative practice problem:

Derivative practice problem 1:

Find the derivative of f(x) = ( x - 5 ) ( 2x + 1 )

Answer:   d/(dx) f(x) = 4x - 9

Derivative practice problem 2:

Differentiate the given equation with respect to t. y =  8t3 + 12t

Answer:  24t2   + 12

Division with Remainders Word Problems

Division is an arithmetic operation which is the opposite of multiplication. The symbol used for division is (÷).

Notation:

Division is often represented by inserting fraction bar which is nothing but the horizontal line between divisor and dividend. Dividend is placed in numerator and divisor are placed in denominator.  For example, a divided by b is written as,

`a/b`

Key terms - division with remainders word problems

Dividend:

The number which is being divided in a division is known as dividend.

Divisor:

The number which divides the dividend is known as divisor.

Quotient:

Quotient is the solution / answer for the problem

Remainder:

The remainder is the amount "left over" after the division

Example problems on division with remainders problems

Division problem 1:

What is division value and remainder of the given fraction 13 / 4

Solution:

When we divide 13 by 4 it gives 3 ( 3 * 4 = 12) as a quotient and 1 is the (13 - 12) remainder value

Quotient = 3

Remainder = 1

Division problem 2:

What is division value and remainder of the given fraction 6 / 4

Solution:

6 / 4 =   There is one 4 in 6

Hence Quotient = 1

Remainder = 2

Solving Division with remainders Word problems:

Division with remainders word problem 1:

A 40 in stick is cut into pieces of 7 in length. How many pieces we can get out of this stick? What is the length of the scrap? Solve the word problem.

Solution:

The problem describes how to divide a 40 in stick into pieces of 7 in each. The remaining piece of the stick is called the scrap.

Therefore, the dividend is 40in., the divisor is 7 in. The quotient of this is the number of pieces and the remainder is the length of the scrap. When we divide 40 by 7, there are fivet 7’s in 40 (5*7= 35) and the remainder is 5.

Hence, the number of pieces of 7 in = 5(quotient)

The length of the scrap = 5 (remainder)

Division with remainders word problem 2:

There are 11 apples and 4 kids . We have to give equal number of apples to each kids. How many apples will each kids get and find the amount of apple left over?

Solution:

Here we have to divide number of apples  10 by number of kids  4.  Hence ,

11 / 4 = 2 with remainder 3

Number of apples each kid will get = 2

The apples left over = 3

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Division with remainders word problem 3:

Consider there are 10 boys in a class. Divide them into 3 groups. How many boys each group will consists and how many boys are left over?

Solution:

Here the dividend is total number of boys 10 and the divisor is 3. We have to divide total number of boys by number of groups to find the number boys each group having .

10 / 3 = 3

Remainder = 1

Hence each group gets 3 boys and 1 boy is left over.

Basic Geometrical Concepts Solve Online

Geometrical is a study of structural mathematics .Basic geometrical has the concept of shapes and structure of an object. It contains two dimensional and three dimensional objects. In this article basic geometrical concepts solve online we are going to see about some basic geometric problems through online.
Generally online help means that here tutor and student should communicate directly with help of online chatting. Student can get more geometrical help and idea from tutor through online with help of chatting. It may be in audio or non audio.


Basic Geometrical Concepts Shapes Solve Online :

Basic geometrical concepts are plane, point, line, angle, ray, two dimensional objects, and three dimensional objects. Main purpose of using geometry is finding area, volume and perimeter of the given shape.

Basic geometrical concepts:

Two dimensional objects:

Square
Rectangle
Triangle
Circle
Parallelogram
Trapezoid

Three dimensional objects:

Cylinder
Cone
Cube
Sphere
Hemisphere
Prism
Pyramid
3d geometry shapes

Solve Example Problems in Basic Geometrical Concepts:

Solve Example problems in basic geometrical concepts:

Example problem 1:

Find the angle of triangle from the given figure?

Geometry-triangle

Solution:

Given data:

Two angles of triangle is 550 and 380

Solve the third angle of triangle

Total angle of triangle=1800

55+38+x=180

93+x=180

X=180-93

X=870

Third angle of triangle is 870

Example problem 2:

Which one is 3D shape of geometrical object?

Geometry- shapes

Solution:

Option (a)-Two dimensional object(square)

Option (b)-Two dimensional objects(Circle)

Option (c)-Three dimensional object(Parallelogram)

Option(d)-Three dimensional object(Cone)

Example problem 3:

How many edges and vertices in the cube?

Cube

Solution:

Cube having 12 edges

Cube having 8 vertices

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So the cube having top portion of four edges

Side portion of four edges

Bottom portion of four edges

So totally have 12 edges

Vertices mean nothing but corner point of the given shapes. When the two edges or lines are meeting It will produce the vertices

Solve Proving Proofs

Mathematical statement considered the sequence of statements by proofs, every statement being adjusted with some definition or a proposition or an axiom that is before launched by the method of deduction using only the permitted logical rules. Repeatedly we prove a proposition straightly from what is given in the proposition. But so many times it is easier to prove a same proposition not proving the proposition by itself,

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Solve Proving Proofs - Types of Proofs:

Solve Proving Proofs Indirect Method:

Solve proving proofs - Indirect method:

Prove the given function f : R  =>  R defined by f(x)  =  2x  +  5 is one – one.

Sol :  A function is one – one if  f(x1)  =  f(x2)  => x1  =  x2.

Using this we have to show that “ 2x1 + 5  = 2x2  +  5” => “  x1  =  x2”.This is of the form p  =>  q, where , p is 2x1  +  5 =  2x2  +  5 and q : x1  =  x2. so this is the direct method proofing.

We can also prove the same by using contrapositive of the statement. Now contrapositive of this statement is ~ q=>  ~ p, is, contrapositive of “if f(x1)  =  f(x2), then  x1  =  x2” is “if x1  =  x2, then f(x1) =  f(x2)”.

Now                 x1  !=   x2

=>                   2x1  !=   2x2

=>                   2x1  +  5  !=   2x2  + 5

=>                   f(x1)  !=   f(x2).
Solve Proving Proofs of Direct Approach:

Solve Proving Proofs -direct Proof:


It  is the proof of  a proposition in which we directly start the proof  with what is given in the proposition.

1)Straight forward approach.

2) Mathematical Induction approach.

3)Proofs By cases or by   exhaustion.

Solve Proving Proofs -Indirect Proof:

Instead of proving the proposition directly, we establish the proof of the proposition through  proving a proposition which is equivalent to the given proposition.

1)Proofs  by contradiction.

2)Proofs by using contrapositive statement.

3)Proofs by a by a counter example.


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Solving Proving Proofs - some Examples for Proving Proofs:

Solving proving proofs - Direct Method:

Prove the given function f:  R  =>  R

Defined by f(x) = 2x + 5 is one – one.

Sol :  Note that a function f is one – one if

f(x1)  =  f(x2)    x1  =  x2 (definition of one – one function)

Now given that   f(x1)  =  f(x2),

=> 2 x1 +  5 = 2 x2 +  5

=>                 2 x1  +  5  –  5  = 2 x2  +  5  –  5(adding the same quantity on both sides)

=>                 2 x1  +  0  =  2 x2  +  0

=>                 2 x1  =  2 x2 (using additive identity of real number)

=>                 (2)/(2) x1  =  (2)/(2) x2 (dividing by the same non zero quantity)

=>                 x1  =  x2.

Sets and Operations

Sets and relations are two of the most important concepts in mathematics that is taught in middle school and is widely used in practical mathematics. A set is a collection of things such as letters, numbers, words, things etc. For example: {woodcraft construction toys, building blocks, construction sets}. This is a set of educational toys such as woodcraft construction toys and so on. Sets also functions different relationship in mathematics. Let’s have a look at the same in this post.

Union of Sets: When two sets are compared and a final set is created with the elements of set A and set B, the set created is called Union of sets. It is referred as A U B. While creating the union of sets, a single element if occurs in both the sets is used only once in the final set.

For example:
A = { Cloth toys for babies , Wooden play toys for kids , Soft toys for babies, Barbie Doll}
B = {Ludo, Chess, Snakes and Ladder, Barbie Doll}
A U B = {

Cloth toys for babies , Wooden play toys for kids, Soft toys for babies, Barbie Doll, Ludo, Chess, Snakes and Ladder}Intersection of Sets: When two sets are compared and a final set is created using the only common elements of set A and set B, then it is called as intersection of sets. It is referred as A ? B.

For example:
A = {Hair, Eye, Nose, Ear, Lips, Hands, Legs}
B = [Hair, Eye, Nose, Ear, Lips, Teeth, Tongue}
A ? B = {Hair, Eye, Nose, Lips}
Difference of Sets: The difference of sets is nothing but the set that includes the unique elements of set A and B. It is referred as A –B. For example:
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8, 10}
A – B = {1, 3, 5, 8, 10}
These are some of the basic relations in sets.

Geometric Ray

We know that, a ray is an important topic of line, and geometrical chapter in mathematics subject. In mathematics, a ray is defined as several ways, when we have seen a vector, a ray vector is going to one point to other point, and it is called as rays. In this geometrical part, ray is used in mostly as the lines and it’s relevant. In this article we are going to explain about the geometric ray, and its definitions.

Definition of the Geometric Ray:

Generally a ray is defined as many ways; it is one part of line.
It has a straight line with one end point and other direction (side) extends to infinity.
And a ray is start with end point, and then other point on the ray next.

For example:

Picture representation for geometric ray:

Geometric rays

If suppose the above ray (figure) is vector ray means, a ray is vector (xy) form of point x to point y.
And a ray is start with end point x, and other side goes to infinity on next. It is named ray.

Note:

If suppose the given ray (see figure) is vector ray means, a ray is vector (xy) form of point x to point y.

Ray is` vec (XY)` .

Exact definition for ray:

Ray is straight line that starts at a point and continues outward in another direction. And a ray is start with end point, and then other point on the ray next.

Geometric rays
Examples Pictures and Explain in Geometric Rays:

Example problem 1: Draw a straight line, with two points ray figure.

Figure:

example figure for ray

Example problem 2: Draw the both opposite side rays.

The ray figures:

example figure for ray, and    example figure for ray

These rays are mostly used in the sphere intersection problems.

Example problem 3: Draw the angle figure used on 2 ray figures with 1 common end point:

Angle figure in ray:

example figure for ray

Example problem 4: Draw the parallel line, and use the ray with intersect (cross) the parallel line.

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ray example

These all above the general explanations and example figures are making clear about the geometric ray definition and concept.

Sequence Listing

In this article, we are going to see about the sequence listing. There are two types of sequences

1. Arithmetic sequence and

2. Geometric sequence.

Arithmetic sequence means, the sequence of numbers such that the difference between two consecutive members of the sequence is a constant. Geometric sequence means, the sequence of numbers such that the ratio between two consecutive members of the sequence is a constant. The sequence listing formulas and example problems are given below.



Formula for arithmetic sequence:

nth term of the sequence : an = a1 + (n - 1)d

Series of the sequence: sn = (n(a_1 + a_n))/2

Formula for geometric sequence:

nth term of the sequence: an = a1 * rn-1

Series of the sequence: sn = (a_1(1-r^n))/(1 - r)
Example Problems for Sequence Listing:

Example problem 1:

Find the 15th term of the series 2, 4, 6, 8, 10,. ....

Solution:

First term of the series, a1 = 2

Difference of two consecutive terms, d = 4 - 2 = 2

n = 15

The formula to find the nth term of an arithmetic series, a_n = a_1 + (n-1)d

So, the 15th term of the series 2, 4, 6, 8, 10....... = 2 + (15 - 1) 2

= 2 + 14 * 2

After simplify this, we get

= 2 + 28

= 30

So, the 15th term of the sequence 2, 4, 6, 8, 10,.... is 30

Series of the arithmetic sequence: sn = (n(a_1 + a_n))/2

 s_15 = [15(2 + 30)]/(2)

 s_15 = [15(32)]/(2)

 s_15 = (480)/(2)

After simplify this, we get

 s_15 = 240



Example problem 2:

Find out the 11th term of a geometric sequence if a1 = 4 and the common ratio (C.R) r = 2

Solution:

Use the formula a_n = a_1 * r^(n-1) that gives the nth term to find a_11 as follows

a_11 = a_1 * r^(11-1)

= 4 * (2)10

= 4 * 1024

After simplify this, we get

= 4096

The 11th term of a geometric sequence is

Series of the sequence: sn = (a_1(1-r^n))/(1 - r)

 s_11 = [4(1 - 2^11)]/(1 - 2)

= [4(-2048)]/(-1)

After simplify this, we get

s_11 = 8192

The above examples are helpful study of sequences listing.