Graphical Interpretation of Derivatives

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 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 the 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 the 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 its original function without knowing the function accurately. We'll discuss these topics here.

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

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

We can see that the graph of a function changes just for changing the sign. And that's all the same case for every function. But, what actually changes? If we look at the functions \(\sin(x)\) and \(-\sin(x),\) 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 \(\left ( \frac{3\pi }{2},-1 \right ).\) 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 the same for both here and they are \(\frac{3\pi }{2}\) and \(\frac{\pi }{2}\).

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

Forget the equation for a moment and just look at the graph. Now, to find the graph of \({f}'\) from the above graph, we have to find two kinds of very important points. We'll put the first kind of points at where \(f'\) is zero or undefined. These places are where the graph turns around and 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 where \(x=-2\) and \(x=2.\) So, in the graph of \({f}'\), we'll plot these 2 points first.

Now, we'll check where the function is increasing and where it is decreasing. For where the function is increasing, we do the following:

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

• Secondly, the function is monotonically decreasing in the interval \(-2< x < 2.\). So, we draw a negative graph for the interval \(-2< x < 2\), which will go towards the negative sides of the \(y\)-axis.

• Thirdly, the function is monotonically increasing in the interval \(2 < x < \infty. \) So, again we draw a positive graph for \(x> 2\), which will go towards the positive sides of the \(y\)-axis.

• At last, the final graph of that function is 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 in more intervals, just plot them as we did above.