# Graphical Interpretation of Derivatives

#### Contents

## Introduction

As we all know, figures and patterns are at the base of mathematics. So, all the terms of mathematics have a graphical representation. If we discuss about derivatives, it actually means the rate of change of some variable with respect to another variable. And , we can take derivatives of any differentiable functions. We can take second , third and more derivatives of a function if possible. When we differentiate a function , we just find out the rate of change. And obsessively the main function has a graph , and when we take derivatives, the graph also changes. If we take second derivative, the graph changes again. This change of a graph due to differentiation follow some rules. We can draw a graph of a differentiated function from it's original function without knowing the function accurately. We'll discuss about these topics here.

## Graphs Of Some Function And Some Points

We know that , if we take the function of \(x^{2}\) we'll get something like smile :

And if we take the graph of \(-x^{2}\) , we just get something opposite of the above that is located at the negative side of x-axis.

We can see that the graph of a function changes just for changing the sign . And there are all the same case for every function. But, what actually changes ? If we look at the function of \(sin\left ( x \right )\) and \(-sin\left ( x \right )\) we can see some changes . Look at the graphs carefully.

These graphs are taken in the interval of \(0\leqslant x\leqslant 2\pi \). For the graph of \(\sin\left ( x \right )\) the local maximum is \(\left ( \frac{\pi }{2},1 \right )\) and the local minimum is \(\left ( \frac{3\pi }{2},-1 \right )\) . And for the graph of \(-\sin\left ( x \right )\) the local maximum is \(\left ( \frac{3\pi }{2},-1 \right )\) and the local minimum is \(\left ( \frac{\pi }{2},1 \right )\) .

Critical Points: a Critical point of a differentiable function is any value into its domain where its derivative is 0 or undefined. The value of that function at a critical point is called the critical value. For the above example , The critical point is same for both here and they are \(\frac{3\pi }{2}\) and \(\frac{\pi }{2}\) .

## Determining The Graph Of Derivative Of A Function

Suppose a function is : \(x^3-12x+3\) and its graph is :

Just Forget the equation for this time. Just look at the graph. Now for finding the graph of \({f}'\) from the above graph ,we have to find 2 very Important Points. We'll put the first points for drawing the graph of \({f}'\) at such a point where the function is ZERO or UNDEFINED . These places are where the graph Turns around. These points are called Turning Points Or Extremes Points Or Critical Points. Here , Looking at the graph , we can see that , The Graph is turned at 2 points. They are \(\left ( -2,2 \right )\). So , at the graph of \({f}'\), we'll plot these 2 points at first.

Now, we'll check where the function is increasing and where it is decreasing. At where the function is increasing ,

We can see that the function is monotonically increasing at the interval of \(-\infty< x < -2\).So , for drawing the graph of \({f}'\) , at first, we'll draw a positive graph for \(x< -2\) , which will go towards the positive sides of X axis. .

Secondly , The function is monotonically decreasing at the interval of \(-2< x < 2\). So , we'll draw a negative graph for the interval \(-2< x < 2\) , which will go towards the negative sides of X axis.

Thirdly , The function is monotonically increasing at the interval of \(2 < x < \infty \). So ,again we'll draw a positive graph for \(x> 2\) , which will go towards the positive sides of X axis.

At last, The Final graph of that function will be something like this :

And you can do the same for any functions. They may be of higher orders. No problem. Just apply the same method. If the function is increasing or decreasing at more intervals, just plot them as above.

How many critical points does the graph has? Find Them And Submit Your Answer As : \[\text{(sum of critical points)} \times \text{(total # of critical points)}-\text{(total # of critical points)}.\]

Just give your answer by observing the graph carefully. If needed, you can zoom in the graph.
This problem includes almost all the favorite number of **Anwesha Biswas Tithi**....................

**Cite as:**Graphical Interpretation of Derivatives.

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