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Derivatives of a Function

Derivatives of a Function

Do you know which is the most difficult mountain to climb? You would probably say Mount Everest, since it is the highest mountain. But it's wrong. The most difficult is the second highest mountain "K 2". Because of the difficulty of its ascent it is called SAVAGE mountain. How to find the minimum value of a function using derivatives.
K2 is more difficult to climb because we know that alongside height there are many other factors to consider. The Steepness is one of them.
For example let’s consider these two mountains. The right one clearly has more altitude. But The left one will be harder to climb as it is steeper. 

For example, consider these two mountains. The one on the right is clearly higher. But the one on the left will be more difficult to climb as it is steeper.
Now let me ask you a question. What do we mean by steepness? 
 For example consider these two straight lines. Which one is steeper? 

Here the difference is subtle, but we can see that it is the second one. But can you think of a way to concretely prove that the second one is steeper? 

One simple way is to draw the horizontal and vertical lines like this, measure their lengths, and find the ratio of the vertical length to the horizontal length. On substituting the values, we get 1 and 1.5 The second ratio is clearly greater. 
What does this tell us? 
If we walk along these lines, this ratio tells us how much VERTICAL distance we cover RELATIVE to the horizontal distance! So if the ratio is higher, it means that the slant line is STEEPER. We can also see this as the distance covered while going up per unit of the distance while going across! There’s another way to know which line is steeper. 
And that is to find this angle. Greater the angle, Steeper will be the line. 
But notice that these two lines are connected to the angle by the trigonometric TANGENT function. This ratio is equal to tan theta one and this is equal to tan theta two. Such a ratio for a straight line is called its slope. It is the measure of the STEEPNESS of the line 
Now look at this curve. What is the slope of this curve? Does it even make sense to ask this? What do you think? 
In this Article, we will understand what this means. Here, the process of differentiation helps us. Also we know that such a curve can be represented algebraically by a function between two variables. We will see that this slope is connected to the idea of the DERIVATIVE of a function
So let’s continue with our question. One thing we can clearly see here is that different portions of the curve will have different steepness. We can intuitively see that as we move from the left towards the right along the curve, it gets steeper. 
So the slope of the curve in general doesn’t make sense. So let’s focus our attention around this point ‘P’. What will be the slope at this point? 
 For this let me ask you one question. Is the Earth flat or round? 
You would probably be laughing because we all know that the earth is almost round in shape. But notice the surroundings around you… it seems that the earth is flat. Why is it so? 
Consider this circle as the representation of the earth. And let’s say you are currently a point on the surface. If we zoom in around this portion of curved line, we see that it becomes less and less curved. It looks almost like a straight line. Let’s say we draw a straight line only intersecting the curve at this point. We can see that the curved line is almost the same as this straight line. We know that this line which passes through only one point of the curve is called the tangent line at that point. 

Take a moment and just observe what we found. This is the most important idea on which Calculus is built. A curved line in a very small region around a point on it, can be approximated by a straight line .This straight line is the TANGENT line at this point. So this solves our problem. The slope of the curve at a point will be equal to the slope of the tangent line at that point. And we’re familiar with how to find the slope of a straight line. 

how do we find the slope of the tangent line at this point? 

But before answering this question, let’s look at WHY we are so interested in finding this slope… We will continue this in the next part. Previously, we saw what we meant by a function between two variables, ‘X’ and ‘Y’. A function tells us how the value of one variable DEPENDS on another variable. So if the value of ‘X’ changes from ‘X not’ to ‘X not plus delta X’ then the value of ‘Y’ will also change from ‘Y not’ to ‘Y not plus delta Y’. 
Now look at this ratio ‘delta Y over delta X’. It tells us the rate at which the value of ‘Y’ changes in proportion to the change in ‘X’. For example, consider these two ratios. As the ratio in the second case is greater, the value of ‘Y’ will change relatively faster here. 
This rate of change for a function is actually related to the slope of a curve. Let us see how. 
Let’s say the graph of this function is a straight line like this. 'X not Y not’ is this point and when the value of ‘X’ changes by ‘delta X’, we reach this point. Now consider the ratio ‘delta Y’ over ‘delta X’. We saw earlier that for a straight line, this ratio is the measure of its steepness. So, the rate of change in ‘Y’ with respect to ‘X’ is equal to the slope of this line. Now we can see that for a straight line, its slope is constant throughout. So its rate of change will also be constant. But what if instead of this, the graph of the function is given by a curved line. 
The slope at a point tells us the rate of change at that point. We saw earlier that in a very small region around a point, the curve can be thought of as a straight line. This is the tangent line at this point. So we see that the slope of this line is equal to the rate of change here. 
But now, we can see that depending on the point on the curve, the slope will change. This means that the rate of change will also vary depending on the value of the 'X' variable. Let us look at an example to understand this. 
Let’s say in this function, ‘Y’ is the distance traveled by an object in motion. And ‘X’ is the time taken for the distance traveled. Then the rate of change in ‘Y’ with respect to ‘X’ will be equal to the speed of the object. 
In the first case, we know that the rate of change is constant. So the speed of the object will be constant throughout its motion. While in the second case, the rate of change is not constant. So here, this rate will tell us the instantaneous speed of the object. 
For this reason the rate of change for a function at a particular value of ‘X’; X not is called the instantaneous rate of change at X not. So to conclude, a very small portion of a curve around a point can be approximated by the tangent line at that point. The slope of this tangent line tells us the instantaneous rate of change of the function at that point. But how do we find this slope or instantaneous rate of change? 

Do you know the instantaneous speed of an object? 

To find the instantaneous speed, we have to use the process of differentiation. 
Let’s say we want to find the instantaneous speed at this point ‘P’, that is at the time instant ‘X not’. 
For this, we first find the average speed for some time interval ‘delta X’. At ‘X not’ the distance covered will be this. And at ‘X not plus delta X’ it will be this. So the average speed in the time interval ‘delta X’ will be equal to this. For a function, this ratio is called the average rate of change. 
Now let’s say we draw a straight line between these two points. It is called the secant line. We see that the slope of this secant line will be equal to the average speed in the time interval ‘delta X’. 
So we see that the average rate of change between two values of ‘X’ is equal to the slope of the secant line between the corresponding points on the curve. 
Now what in the next step? How do we find the instantaneous speed from this average speed? For this we find the average speed in shorter time intervals, that is, as ‘delta X’ tends to zero. 
We say that when the limit ‘delta X’ tends to zero, the average speed approaches the instantaneous speed. . 
This instantaneous speed is the instantaneous rate of change at ‘X not’. Now what happens to this secant line as ‘delta X’ tends to zero? 
Let’s say we have to find the average rate of change between points ‘P’ and ‘Q one’, ‘Q two’ and so on. We can see that as ‘delta X’ tends to zero, these points come closer and closer to the point 'P'. So as the limit ‘delta X’ tends to zero, the secant line will pass through only one point – ‘P’ on the curve. That is the secant line approaches the tangent line. 
This is how using the process of differentiation we find the instantaneous rate of change of a function or the slope of a tangent line. Now here instead of writing the instantaneous rate of change in ‘Y’ with respect to ‘X’ at ‘X not’, we say that the derivative of the function at ‘X not’ is this.The derivative is usually denoted by the symbol ‘D Y by D X’. 
So we saw here that the derivative of the function at a particular value of ‘X’ is its instantaneous rate of change at that point. 

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