I am going to explain what is differentiation. I will explain it with a real life example. Suppose you are driving a car and you are climbing a small hillock, but you have put second gear all along and constant push in the pedal. I want to know what is the acceleration of the car at any given point. Speed changes as you move along, because when you climb the hill, speed is going to decrease, because gravity pulls the vehicle. When you reach the top and climb down with the same gear, the speed is going to increase, again because gravity pulls you. We will see it in a diagram
Suppose the car starts at the speed of 9 Kilometres/Hour ( For convinience ) from the foot of hillock and in 3 hours it reaches the top of the hillock. Now what differentiation is doing here. Differentiation helps you to find the acceleration of the vehicle at any point in the journey. Acceleration is nothing but how rapidly the speed of vehicle changes.
Now we will get into mathematics,
At point A, suppose the car starts at a speed of 9 kmph, which is at the bottom of the hillock. Supposing we are giving a certain push in the accelerator, speed keeps on decreasing and at point B, the car has stopped, we assume that it stops at the top of the hillock. After that, the speed starts increasing, because gravity increases its speed. If we put this graph in a mathematical equation, it would be y=(x-3)^2, where y is speed and x is time. The acceleration of the vehicle at any point is the slope of a tangent on that particular point.
Suppose, we want to calculate the acceleration of the vehicle after 2 hours, we have to draw a tangent at the intersection of x = 2 and the parabola. (Tangent is a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point. Parabola is name of the curved graph.) The slope of that tangent is the acceleration of the car at that point. It is tedious to find the acceleration of the vehicle by drawing tangents and graphs. So guess, who solves it. Differentiation. If we differentiate an equation like this parabola, it will give a general formula, which will tells us the acceleration of the car( change in the speed of the car) at any point in time.
More mathematics,
We will study the graph, at Point E, the slope of the tangent to the graph is high and when it reaches the point C, the slope becomes lesser and lesser and it becomes 0 at point B. Again it starts increasing.
Now we will try to differentiate this graph y = (x-3)^2 using graphic method, to actually understand why we get the differentiation formulas. If we differentiate this parabola, what we get is the general formula which tells about slope( acceleration) of the parabola at every point, as x ( Time) changes. We will plot three points in the parabola and find the slope at that three points.
We know that the slope of the tangent decreases slowly from point E to point B, where it becomes 0. So, there must be some point in the parabola, between point E and B, where the slope of the curve becomes 1. Plot a triangle, such that the adjacent side ( Time axis ) and opposite sides ( Speed axis ) are equal, and the hypotenuse side ( Tangent ) touches exactly at one point in the parabola. By trial and error method, we can find the point at which x and y are equal and solves the parabola curve. i.e if we substitute x and y in the equation y = ( x - 3 )^2, the equation solves. Since the opposite and adjacent sides are equal, the slope of the hypotenuse side is exactly 1. Since hypotenuse side touches at only one point in the parabola, it becomes the tangent to the curve at that point.
If we plot a line with x axis as time and y axis as slope of the tangent of the parabola at that point, we will get the following equation y = 2x - 6. [ Note : We can find that from equation y = mx + b, where m is slope. Since we know 3 points in the line, we can form the equation. ]
If we differentiate y = ( x - 3 )^2 , we get the same equation dy/dx = 2x - 6. So differentiation is nothing but rate of change. Why do i need a rate of change of speed ( acceleration ), because it will come handy in lot of places like launching a rocket. we need to know and adjust acceleration to put our satellites exactly in the orbit.
We will get into sine curve
If we plot rate of change of sine curve ( i.e slope of tangent to the curve ), naturally we get a curve and it happens to be cosine curve.
Suppose the car starts at the speed of 9 Kilometres/Hour ( For convinience ) from the foot of hillock and in 3 hours it reaches the top of the hillock. Now what differentiation is doing here. Differentiation helps you to find the acceleration of the vehicle at any point in the journey. Acceleration is nothing but how rapidly the speed of vehicle changes.
Now we will get into mathematics,
At point A, suppose the car starts at a speed of 9 kmph, which is at the bottom of the hillock. Supposing we are giving a certain push in the accelerator, speed keeps on decreasing and at point B, the car has stopped, we assume that it stops at the top of the hillock. After that, the speed starts increasing, because gravity increases its speed. If we put this graph in a mathematical equation, it would be y=(x-3)^2, where y is speed and x is time. The acceleration of the vehicle at any point is the slope of a tangent on that particular point.
Suppose, we want to calculate the acceleration of the vehicle after 2 hours, we have to draw a tangent at the intersection of x = 2 and the parabola. (Tangent is a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point. Parabola is name of the curved graph.) The slope of that tangent is the acceleration of the car at that point. It is tedious to find the acceleration of the vehicle by drawing tangents and graphs. So guess, who solves it. Differentiation. If we differentiate an equation like this parabola, it will give a general formula, which will tells us the acceleration of the car( change in the speed of the car) at any point in time.
More mathematics,
We will study the graph, at Point E, the slope of the tangent to the graph is high and when it reaches the point C, the slope becomes lesser and lesser and it becomes 0 at point B. Again it starts increasing.
Now we will try to differentiate this graph y = (x-3)^2 using graphic method, to actually understand why we get the differentiation formulas. If we differentiate this parabola, what we get is the general formula which tells about slope( acceleration) of the parabola at every point, as x ( Time) changes. We will plot three points in the parabola and find the slope at that three points.
We know that the slope of the tangent decreases slowly from point E to point B, where it becomes 0. So, there must be some point in the parabola, between point E and B, where the slope of the curve becomes 1. Plot a triangle, such that the adjacent side ( Time axis ) and opposite sides ( Speed axis ) are equal, and the hypotenuse side ( Tangent ) touches exactly at one point in the parabola. By trial and error method, we can find the point at which x and y are equal and solves the parabola curve. i.e if we substitute x and y in the equation y = ( x - 3 )^2, the equation solves. Since the opposite and adjacent sides are equal, the slope of the hypotenuse side is exactly 1. Since hypotenuse side touches at only one point in the parabola, it becomes the tangent to the curve at that point.
If we plot a line with x axis as time and y axis as slope of the tangent of the parabola at that point, we will get the following equation y = 2x - 6. [ Note : We can find that from equation y = mx + b, where m is slope. Since we know 3 points in the line, we can form the equation. ]
If we differentiate y = ( x - 3 )^2 , we get the same equation dy/dx = 2x - 6. So differentiation is nothing but rate of change. Why do i need a rate of change of speed ( acceleration ), because it will come handy in lot of places like launching a rocket. we need to know and adjust acceleration to put our satellites exactly in the orbit.
We will get into sine curve
If we plot rate of change of sine curve ( i.e slope of tangent to the curve ), naturally we get a curve and it happens to be cosine curve.
