Derivative Calculator Step by Step

In calculus, the rate of change of curve is called derivative. Derivative Calculator is used to find the differentiation of given function. By Using the derivative calculator we can able to find the derivative value easily. Through manual calculator we can able to solve the derivative but it quite difficult to perform the calculation. But through calculator there is no other difficulty rather than entering the question, the answer will produced automatically. we have to enter the question in function place. then enter the value in which we derivative with respect to. then the order of the respected derivative, now click the derivative button. sample derivative calculator is given.

Derivative Calculus Formulas - Derivative Calculator Step by Step:

1. n th power derivative is `d/dx` (xn) = n x(n-1)

2. Exponential derivative `d/dx` (ex) = ex

3. Logarithmic derivative `d/dx` (log x) = `1/x` .

4. Trigonometry derivative `d/dx` (sin x) = cos x

5.Trigonometry derivative `d/dx` (cos x) = - sin x

6.Trigonometry derivative `d/dx` (tan x) = sec2x

7.Trigonometry derivative `d/dx` (sec x) = sec x tan x

8.Exponential derivative `d/dx` (eax) = aeax

9.Product derivative `d/dx` (uv) = u` (dv)/(dx)` + v `(du)/(dx)` .

10.Division derivative `d/dx(u/v)` = ` [v (du)/(dx) - u (dv)/(dx)]/v^2` .

Example Problems - Derivative Calculator Step by Step:

Derivative calculator step by step problem 1:

Find the derivative of given function      x6 + 3x4 + 5x2 + 3x.

Solution:

Given function is    x6 + 3x4 + 5x2 + 3x.

Step 1:      The derivative of given function can be expressed as  `d/dx` (x6 + 3x4 + 5x2 + 3x).

Step 2:      Separate the derivative function is   `d/dx` (x6) + `d/dx`(3x4) + `d/dx` (5x2 ) + `d/dx`(3x).

Step 3:      Now we know the n th power derivative rule. That is `d/dx` (xn) = n x(n-1)

Step 4:       In the first term n = 6, So derivative of   `d/dx` (x6) = 6 x(6-1)  = 6x5

Step 5:    Differentiate all the term, we get    =   `d/dx` (x6) + `d/dx`(3x4) + `d/dx` (5x2 ) + `d/dx`(3x).

=  6 x(6-1)  +  (3)(4) x(4-1)  + (5)(2) x(2-1)  + (3)(1) x(1-1)

=  6x5 + 12x3 + 10 x + 3

Answer:  6x5 + 12x3 + 10 x + 3.


Derivative calculator step by step problem 2:

Find the derivative of given function       5x2 -  3tanx .

Solution:

Given function is   5x2 -  3tanx .

Step 1:      The derivative of given function can be expressed as  `d/dx` (5x2 -  3tanx ).

Step 2:      Separate the derivative function is    `d/dx` (5x2 ) - `d/dx`(3 tan x).

Step 3:      Now we know the n th power derivative rule. That is `d/dx` (xn) = n x(n-1)

Step 4:       In the first term n = 2, So derivative of   `d/dx` (x2) = 2 x(2-1)  = 2x

Step 5:    Differentiate all the term, we get    =   `d/dx` (x2) - `d/dx`(3 tan x).

=  2x  - 3 sec2x .

=  2x  - 3 sec2x .

Answer:   2x  - 3 sec2x .

Study Online Limits of Trigonometric Functions

In online, trigonometric function are recognized as circular function of an angle. They are organizing to transmit the position of a triangle toward the lengths of the surface of a triangle. Let x, an angle in radians `x->0` `sin x->0`, so `lim_ (x->0)(sinx)` is of the in-between type of type 0. In online we acquire more information regarding all the problems contain step by step solution. Various websites are present in study online limits of trigonometric functions.

Study Online Limits of Trigonometric Functions:

The general rule of limit trigonometric function:

`"lim_(x->a) `  =>    `lim_(x->a) f(x) =0 `

Trigonometric functions acquire in a large variety of uses to include found indefinite lengths as well angles in triangles often right triangles.

Function of an angle expressed though ratios of the length of the surface of right-angled triangle contain the angle.

Trigonometric function is the proportion of the exterior of the right triangle. Function of an angle articulated while ratios of the length of the sides of right-angled triangle have the angle



Trigonometric function:

Sin = `(opposite)/("hypotenuse")`
Cos =` (adjacent)/("hypotenuse")`
tan =`(opposite) / (adjacent)`
cot  = `(adjacent) / (opposite)`
sec = `("hypotenuse") / (adjacent)`
csc =`("hypotenuse")/(opposite)`

There are three reciprocal function in trigonometric.

Csc x = `1/(sinx) `

Sec x =`1/(cosx) `

cot x =`1/(tanx) `

Examples for Study Online Limits of Trigonometric Functions:

Example 1 for  study online limits of trigonometric functions:

How to find `lim_(x->o) (sin 8x)/(8x)`

Solution:

Step 1: the given function is  `lim_(x->o) (sin 8x)/(8x)`

Step 2: let t = 4x we have `t->0` when `x->0`

Step 3:  `lim_(x->o) (sin 8x)/(8x)`

=    `lim_(t->o) (sin t)/(t)`

= 1.

So the solution is     `lim_(x->o) (sin 8x)/(8x)`      = 1.

Example 2 for study online limits of trigonometric functions:

How to find  `lim_(x->4) (sin (x-4))/(x^2-9x+20)`

Solution:

Step 1: The given function is `lim_(x->4) (sin (x-4))/(x^2-9x+20)`

Step 2:   `lim_(x->4) (sin (x-4))/(x^2-9x+20)`

`lim_(x->4) (sin (x-4))/((x-5)(x-4))`

Step 3:     `lim_(x->4)(sin(x-4)/(x-4)*1/(x-5))`

Step 4:  `1*-1 =-1`

So the solution is    `lim_(x->4) (sin (x-4))/(x^2-9x+20) = -1`

Example 3 for  study online limits of trigonometric functions:

How to find `lim_(x->pi/3) (sin x)`

Solution:

Step 1: the given function is `lim_(x->pi/3) (sin x)`

Step 2:   `lim_(x->pi/3) (sin x)` <br>

Step 3:   = 0.8660

So the solution is     `lim_(x->pi/3) (sin x) = 0.8660`

Example 4 for study online limits of trigonometric functions:

How to find  `lim_(x->-4) (sin (x+4))/(x^2+6x+8)`

Solution:

Step 1: The given function is   `lim_(x->-4) (sin (x+4))/(x^2+6x+8)`

Step 2:   `lim_(x->-4) (sin (x+4))/(x^2+6x+8)`

`lim_(x->-4) (sin (x+4))/((x+2)(x+4))`

Step 3:     `lim_(x->-4)(sin(x+4)/(x+4)*1/(x+2))`

Step 4:  `1*-1/2 `

Step 5:      =   `-1/2`

So the solution is   `-1/2`

Absolute Property Solutions

Basically absolute values mean a value without considering its sign. Here we are going to find the solutions of absolute properties. If we are having any number x mean absolute value of x is denoted like |x| and its value is +x. Likewise for |-x| = x. Here we are going to learn the properties of the absolute value and its solutions. If we know the properties of the absolute value it is easy to do the operations on the absolute values.

Absolute Property Solutions:

The first property is non negative property.

Absolute property solutions - Non negativity property:

Absolute value of the numbers is always greater than 0. if we are having a negative number its absolute value is a positive number.

Example:

What is the absolute value of the number -1.2?

Solution:

The given number is -1.2

We have to find the absolute value of the number -1.2

So |-1.2| = +1

So always |x| > 0

Absolute property solutions - Positive definiteness:

The second property is positive definiteness.

The absolute value of the number 0 is always 0. If |x| = 0 then x = 0 (always)

Example:

What is the absolute value of 0?

Solution:

Basically an absolute value is having two values. But for 0 the absolute value is |0| = 0

Absolute property solutions - Multiplicative property:

Multiplication of any two given absolute values equals to the individual absolute value multiplication.

|x y | = |x| |y|

Example:

|-5 x 3| = |-15| = + 15

|-5| x |3| = +5 x +3 = + 15

So both the values are same.

More Absolute Property Solutions:

Absolute property solutions - Subtraction addition property:

Addition of any two given absolute values is always less than the value of its individual addition.

|x + y| ≤ |x| + |y|

Example:

|4 + -3| = |1| = 1

|4| + |-3| = 4 + 3 = 7

1 < 7

Absolute property solutions - Symmetry property:

Symmetry property mean |-x| = |x|

Absolute value of the –x and absolute value of x is always same.

Example:

|-4| = |4| = +4

Absolute property solutions - Identity of indiscernible property:

The difference of any two absolute values of the number is 0 then these two absolute values are same.

|x - y| = 0 then x = y

Example:

Find the absolute value of |5 – (-5)|

Solution:

Absolute value of |5| = +5

Absolute value of |-5| = +5

So |5 – (-5)| = 0

Absolute property solutions - Preservation of division:

The division of any two absolute values and its individual absolute value division are equal.

|x / y | = |x| / |y| (where y ≠0)

Decimal to Percentage Conversion

The conversion of one model to another is the most important thing in the world, which one can never
deny. The above sentence is so assertive because of the underlying fact that learning is important at
every aspect, since it is important for the take away we get out of something. Conversion basically gives
a general perception that it is usually simplified to a simpler form to get a better understanding, so it is a
positive word which gives a positive meaning. This concept is all the same like above, decimal or fraction
conversion into the other.

The process of  Decimal to Percent conversion is done when one is good in basics and fundamentals of
mathematics. First of all one has to be good in decimal which will help in identification of the number
when posed for solving. Decimal can also be called as fraction, when a fraction is simplified the result
will be a decimal. The above will help in solving the question Convert Decimal to Percent. The process of
conversion is not rocket science, it is as simple as solving the basic addition and subtraction.

Converting Decimals to Percents basically involves basic multiplication. The multiplication of the decimal for the conversion is 100 and the symbol used for percent-age is %. This symbol signifies the percentage.

This conversion can be explained clearly using the following example; the question will be given as
convert 0.6 Decimal to Percentage, the answer for the question will be 60%, which is simplified by
multiplying the number 0.6 by 100. The answer will be 60% when multiplied. This can be further
explained by considering another example, convert 0.75 Decimals to Percents, the answer for this
example will 75%, which is resulted after multiplying 0.75 by 100.

Now it is time to think about some of the real life scenarios, where we will need this concept for
applying. Of course every single thing that one studies or learnt should be applied sometime or
somewhere at some point of his / her life, if one does not uses or applies it, there is no point in learning
things. It is mostly used while forecasting the sales of any firm because it is always essential to use
percentile rather than using decimals while portraying the results, which will make the one to whom the
results are presented understand the numbers which resulted.

Intersection of Two Straight Lines

Two lines will intersect at a point. The point will have a pair of values as (x1, y1). The straight lines are represented by equation with two or one variable in x and y.  If those equations are solved to get the value of x and y, will represent the point intersection of those two lines. To solve the set of lines to get the value of x and y , we can use either the method of elimination or substitution. Now let us discuss few problems on this topic intersection of two straight lines.

Example Problems on Intersection of Two Straight Lines

Ex 1: Find the point of intersection of the following two lines

2x + 3y = 10; 2x + y = 6.

Sol: Given: 2x + 3y = 10 --------------(1)

2x + y = 6  --------------(2)

The point intersection of above two lines can be obtained by solving them as follows:

(1) – (2) implies: 2x + 3y = 10

2x + y = 6

We get, 2y = 4

y = 2.

From (2), 2x + (2) = 6

2x = 6 – 2 = 4

2x = 4

x = 2.

Therefore, the point of intersection is (2, 2).

Ex 2: Find the point of intersection of the following two lines

x + 2y = 1; 5x + 4y = -7.

Sol: Given: x + 2y = 1 ------------------(1)

5x + 4y = -7 -----------------(2)

The point of intersection of above two lines can be obtained by solving the as follows:

(1) x 5 – (2) implies: 5x + 10y = 5

5x + 4y = -7

We get, 6y = 12

y = 2.

Therefore, from (1), x + 2(2) = 1

Implies, x = 1 – 4 = -3.

Therefore the point of intersection is (-3, 2).

Ex 3: Find the point of intersection of the following two lines

3x + y = 10; y = 7.

Sol: Given: 3x + y = 10 -------------(1)

y = 7 -----------(2)

Since, y = 7 is one of the line, the value of the y coordinate will be 7.

Therefore, from (1), we get , 3x + 7 = 10

3x = 10 – 7 = 3

x = 1.

Therefore, the point of intersection is ( 1, 7).

Practice Problems on Intersection of Two Straight Lines

1. Find the point of intersection of lines 3x + 2y = 7 and x + y = 3.

[ Answer: (1, 2)]

2. Find the point of intersection of lines 3x - 2y = -2 and x - 2y = 6.

[ Answer: (-4, -5)]

3. Find the point of intersection of lines 4x - 3y = -10 and 3x + y = -1.

[ Answer: (-1, 2)]

Solving Binomials by Factoring

Introduction: 

In algebra, the polynomials which have two terms are called binomials. To factor binomials, we need to follow the following methods:

(i) 2a + ab = a(2 + b ) [ Here the given expression has two terms, where a is the common value]    
     = a( 2 + b)         
(ii) a² – b² = ( a + b) ( a – b)   [ This is the standard form]
(iii) (a + b)² = ( a + b)(a +b)
(iv) (a - b)² = ( a - b)(a - b)

Product of two polynomials will give three terms.
( a + b)² = ) a² +b² + 2ab
( a - b)² = ) a² +b² - 2ab

Let us see few problems on this topic solving binomials by factoring.

Example Problems on Solving Binomials by Factoring

Ex 1: Solve (x + 3) (x – 4) = 0

Soln: Given: (x + 3) (x – 4) = 0
This implies: x + 3 = 0 or x – 4 = 0
That is : x = -3, 4.
Therefore the solution is { -3, 4}.

Ex 2: Solve x + y = 7 and xy = 12, find x and y.

Soln: Given : x + y = 7 -----------(1)
                            xy = 12 ---------(2)
Therefore, x – y = sqrt [(x+y)² – 4xy]
                             = sqrt[ 7² – 4(12)]
                             = 1
Therefore, x – y = 1 ---------------(3)
From (1) and (3), we get:
x + y = 7 -----------(1)
x – y = 1 -----------(3)
2x = 8, this implies that x = 4.
Therefore, (1) implies 4 + y = 7
Hence y =3.
Therefore the solution is {4,3}.

Ex 3: Solve x - y = 5 and xy = 24, find the value of x + y.

Soln: x + y = sqrt [(x-y)² + 4xy]
                   = sqrt[ 5² – 4(24)]
                   = 11.
Therefore from, x + y = 11
                             x – y = 5,
We get 2x = 16.
Therefore, x = 8.
Hence from x + y = 11. y = 3.
Therefore the solution is { 8, 3}.

Ex 4: Solve x + y = 11 and xy = 24, find the value of x² – y².

Soln: x – y = sqrt [(x+y)² – 4xy]
                   = sqrt[ 11² – 4(24)]
                   = 5
Therefore, x² – y² = ( x + y )( x – y)
                             = 11 x 5 = 55.

Practice Problems on Solving Binomials by Factoring

1. If a + b = 9  and ab = 36, find a - b
[Ans: a – b = 5]

2. If a – b = 4 and ab = 12, find a² – b².
[Ans: a² – b² = 32]

Graph of Sinx

 The coordinate graph is called the Cartesian coordinate plane. The graph contains a couple of the vertical lines are called coordinate axes. The vertical axis of the y axis value and the horizontal axis value is the x axis value. The points of the intersection of those two axes values are called the origin of coordinate graphing pictures.  The trigonometry graph is a sin or cos waves. In this graph equation is in the form of y = mx + c. m is nothing but a sin or cos. In this article we shall discuss graph of sin x.

Sample Problem for Graph of Sin X:

Graph of sin x problem 1:

Solve the given trigonometry functions 3sin 5x - y = 0 and draw the graph for the given function.

Solution:
In the first step we find the plotting point of the given trigonometry functions. The given function is
                                      3sin 5x - y = 0
We are going to find out the plotting points for a given equation. In the first step we are going to change equation in the form of y = mx + c, we get the following term
                                  3sin 5x – y = 0
                                                y = 3sin 5x
In the next step we are find out the plotting points of the above equation.
In the above equation we put x = -5 we get
           y = 3sin 5(-5)
           y = -0.4
In the above equation we put x = -4 we get
            y = 3sin 5(-4)    
           y = -2.7
In the above equation we put x = 0 we get
            y = 3sin 5(0)
           y = 0
In the above equation we put x = 2 we get
            y = 3sin 5(2)
           y = -1.6
From equation (1) we get the following value

X	-5	-4	0	2
y	0.4	-2.7	0	-1.6

Graph:

Graph of Sin X Problem 2:

Solve the given trigonometry functions y = 2sin 3x and the draw graph for the given function.
Solution:
In the first step we find the plotting point of the given trigonometry functions. The given function is
                      y = 2sin 3x
We are going to find out the plotting points for a given equation. In the first step we are going to change equation in the form of y = mx + c, we get the following term
                        2sin 3x – y = 0
                              2sin 3x = y                                   
In the above equation we put x = -4 we get
           y = 2sin 3(-4)
           y = 1
In the above equation we put x = -3 we get
            y = 2sin 3(-3)   
           y = -0.82
In the above equation we put x = 0 we get
            y = 2sin 0
           y = 0
In the above equation we put x = 3 we get
            y = 2sin 3(3)
           y = 0.82
From equation (1) we get the following value

x	-4	-3	0	3
y	1	-0.82	0	0.82

Graph: