Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Learn Linear Functions Slope

A linear function is defined as the polynomial function contains the degree of one (y = mx + b). One can learn the linear equation relates a dependent variable with an independent variable in a simple way. The power of the linear function which is not always greater than one where there is no independent variable. A simple linear function with one independent variable (Ax + By + C = 0) traces a straight line when plotted on a graph. It is also called as linear equation. Learning the concept of slope using linear equations is known as learning linear functions slope.

Please express your views of this topic Linear Approximation Equation by commenting on blog.

Learning Linear Functions Forms:

The function is defined by,

f = { ( X, Y)/ Y = mX + b }

where m and b are constants, x and y is called a linear functions. The function derives a straight line while graphing.

Functions such as these gives graph that are straight lines, and, thus, the name linear. Linear functions come in three main forms.

Point Slope Form is given by the equation, m = (y - y1) / ( x – x1)
Slope-Intercept Form is given by the equation, y = mx +b
General Form is given by the equation, Ax + By + C = 0.


Slope of the Linear Functions Learning:

Calculations of rate at which the change takes place can be done under the concept of slope. Slope calculates the rate of change in the dependent variable as the independent variable changes. The slope is denoted by m.

Consider the linear function:

y = mx + b

where, m is the slope of the line and b is the y-intercept. Slope is defined as the ratio of unit change in y to the change in x.

slope m = Change in y / Change in x

=> m = (y_(2) - y_(1))/(x_(2) - x_(1))

Learning Linear Function Slope - Examples:

Find the slope of the line segment relating the following points:
(-1,-2) and (1, 6)

Sol:

Here, x1 = -1        

y1 = -2

x2 = 1        

y2 = 6

slope m = (y2 – y1) / (x2 – x1)

=> m = (6 – (-2) / (1 – (-1))

=> m = (6 + 2) / (1 + 1)

=> m = 8 / 2

=> Slope m = 4


Find the slope of the equation, 9x - 3y = 6

Sol:

9x - 3y = 6

=> -3y = 6 – 9x

=> y = (-1/3)(6 – 9x)

=> y = 3x – 2

It is in the general form, y = mx + b

Therefore, slope m = 3 and y-intercept b = -2.

Continuity Learning

Let 'a' be an aggregate and a in A, A function f : A |-> R is said to be continuous at "A". If each E > 0 EE   delta >0 such that x in A, || < delta  x - a

rArr   |  f ( x ) - f ( a ) | < E
lim_(x->a) f ( x ) = f ( a )
f is continuous at a    hArr lim_(x->a) f ( x ) = f ( a )

A function f : A |-> Ris said to be continuous on the aggregate 'a'. If ' f ' is continuous at every point a in A
A function f : A -> R is said to be discontinuous at a A. If ' f ' is not continuous at ' a '.


Continuity on the left and right at 'a':-

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a - delta < x <= a  rArr | f ( x ) - f ( a ) | < E

lim_(x->a-)  f ( x ) = f ( a )

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a <= x < a + delta   rArr | f ( x ) - f ( a ) | < E

lim_(x->a+) f ( x ) = f ( a )

Continuity of a function at open and closed intervals:-

A function ' f ' is said to be continuous on an open interval ( a, b ). If ' f ' is continuous at ' x ' AA x in ( a, b ).

A function ' f ' is said to be continuous on an closed interval [ a, b ].

' f ' is continuous at 'x' AA x in ( a, b )
' f ' is right continuous at 'a'.
' f ' is left continuous at ' b'.


Types of discontinuity:-

Let f : A -> Rand a in A be a point of discontinuity of ' f '.

' f ' is said to have Removable discontinuity at 'a'. If lim_(x->a) f ( x ) exists and lim_(x->a) f ( x ) != f ( a ) or f ( a )  is not defined.
' f ' is said to have Jump discontinuity or discontinuity of first kind at 'a'. If lim_(x->a-) f ( x ), lim_(x->a+) f ( x ) both exists and

lim_(x->a-)    f ( x )        !=  lim_(x->a+) f ( x )

' f ' is said to have sample discontinuity at 'a'. If ' f ' has removable discontinuity ( or ) jump discontinuity at 'a'.
' f ' is said to have finite discontinuity at 'a'. If ' f ' is not continuous at 'a' and ' f ' is bounded at 'a'.
' f ' is said to have Infinite discontinuity at 'a'.

If lim_(x->a) f ( x ) = oo   ( or ) lim_(x->a) f ( x ) = -oo
f ( x ) is unbounded in every neighbourhood of 'a'.

Example problems on Continuity

1)Examining the continuity of ' f ' defined by f ( x ) = | x | + | x - 1|

Solution:-    | x| = x if x > 0
= - x if x <= 0
| x - 1| = x if x > 1
= - x if x <= 1

If x <= 0
f ( x ) = | x | + | x - 1|
= - x - ( x -1 )
= 1 - 2x

If 0 < x < 1
f ( x ) =  | x | + | x - 1|
= x  - ( x - 1 )
= x - x + 1
= 1

If x >= 1
f ( x ) = | x | + | x - 1|
= x + x -1
= 2x -1

f ( x ) = 1 - 2x ; If x <= 0
= 1        ;  if 0 < x < 1
= 2x -1  ; if x >= 1

Continuity of ' f ' at x =0

Left Hand Limit            lim_(x->0-) f ( x ) = lim_(x->0-) 1 - 2x = 1

Right Hand Limit          lim_(x->0+) f ( x ) = lim_(x->0+) 1  = 1

f ( x ) = 1 - 2x if x <= 0
f ( 0 ) = 1

lim_(x->0) f ( x ) = f ( 0 ) = 1

Continuity of ' f ' at x =1

Left Hand Limit             lim_(x->1-) f ( x ) = lim_(x->1-) 1  = 1

Right Hand Limit            lim_(x->1+) f ( x ) = lim_(x->1+) 2x - 1  = 1

f ( x ) = 2x -1 if x >= 1
f ( 1 ) = 2 ( 1 )  - 1 = 1

lim_(x->1) f ( x ) = f ( 1 ) = 1

The function is continuous at x = 1

Intersecting Acute Angles

The intersecting acute angles measure the 0° and 90°. A sketch of angles is the main topic in geometry. Two lines join at a single point to form an angle.

If P and R are two straight lines they intersect at a point Q and make an angle Q. The angle Q is also represented as PQR. Here center point represents the vertex of the angle.

Example problems for intersecting acute angles:

Problem 1:-

Solving the intersecting acute angle from the where slopes  m1= 4 and m2=14

Solution:

In above diagram show the acute angles intersecting the two lines

tan phi = (m2- m1)/ (1+m1m2)

= (14-4)/(1+(14x4))

=10/ 57

= 0.175

Hence we find the phi value.

phi = arc tan(0.175)

from the tangent table,we get
phi =1o  (rounded value)



from the given example we can find the obtuse angle.

To subtract the acute angle between the the two values L,L2 from the 1800 straight angle.

That is,

phi = 180- phi

phi = 180 -1o

phi = 179o

Hence we find the obtuse angle.

Problem 2:-

Solving the intersecting acute angle from the where slopes  m1= 20 and m2=28

Solution:

In above diagram show the acute angles intersecting the two lines

tan phi = (m2- m1)/(1+m1m2)

= (28-20)/(1+(28x20)

= 8/ 561

= 0.0142

Hence we find the phi value.

phi   = arc tan(0.0142)

from the tangent table,we get
phi   = 1o  (rounded value)

from the given example we can find the obtuse angle.

the obtuse angle is calculated by subtracting the acute angle between the value of L and L2 from the straight angle 180 0

ie phi = 180- phi

phi = 180 -1o

phi = 179o

Hence we find the obtuse angle.

My forthcoming post is on algebra 2 formulas sheet and sample papers for class 10 cbse will give you more understanding about Algebra.

Practice problems for intersecting acute angles:

1. Solving the intersecting acute angle from the where slopes  m1= 2 and m2=4

Answer:- 168o

2. Solving the intersecting acute angle from the where slopes  m1= 4 and m2= 8

Answer:- 173o

3. Solving the intersecting acute angle from the where slopes  m1= 8 and m2=12

Answer:- 178 o

4. Solving the intersecting acute angle from the where slopes  m1= 12 and m2=14

Answer:- 174 o

Time Value of Money

Time value of money is one of the most important concepts in mathematics, also widely used in real life finance and economical scenarios. Time value of money is taught sometime in high school mathematics that plays a major role in the practical life. Let’s discuss about the same in this post.

What is Time Value of Money or TVM?
Time value of money is popularly abbreviated as TVM. The concept of Time value of money defines that the value of money keeps changing with time. Most of the time, the value of money today is lesser than the value of money tomorrow. For example: The popular parenting magazine cost was Rs.50 till last year. Today, the cost of the same parenting magazine is Rs.75. This demonstrates that in the period of one year, the value of money has changed. Understanding the concept of Time value of money requires understanding in related concepts like Present value and Future value.Present Value and Example:

The current worth of money is termed as the present value. For example: Only for today they have offered the scheme of Rs.699 for post pregnancy weight loss program. From tomorrow the rate of post pregnancy weight loss program will be again Rs.1799. Therefore, the present value here is Rs.699. Based on the present value and the difference in future, one can find out the rate of increase or decrease in TVM.

Future Value and Example:
Future value is the amount or value of money on a specified date in future with respect to the same in today’s date. For example: The book on develop reading habit cost is Rs.100 today but the cost of same book on develop read habit will increase to Rs.150 by next month, owing to its popularity. Thus, Rs.150 is the future value in this situation.
These are the basics on Time Value of Money.