Derivatives

In the 17th century, Fermat wanted to find the lowest (minimum) and highest (maximum) points of a curve within a certain range. Fermat noticed that at these points, the tangent lines were horizontal, and therefore the slope of the tangent line at these points is zero.

In that way, Fermat noticed that the problem of finding maximum and minimum points is related to the problem of finding tangent lines.

How to calculate tangent lines

Consider the following graph:

In the image above we have represented a graph of function $f(x)$ in which is highlighted the point $(a,f(a))$. The goal is to calculate the tangent line at this specific point — the red line.

Every line can be represented with the following equation:

$$y=mx+b$$

in which:

• $y$ is the ordinate of the point of the line
• $x$ is the abscissa of the point of the line
• $m$ is the slope of the line
• $b$ is the y-intercept of the line

$m$ is defined by the equation of the tangent of the angle that certain line makes with the x-axis, therefore $m=\tan (\theta )$.

Given this definition, it is possible to calculate $m$ from two points in the line with the following equation:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

But we do not have two points, we only have one point and we have to find out its slope. A way to think is to consider a secant line in the function graph:

In the above graph we added the point $(b,f(b))$ and drawn a secant line passing through the points $(a,f(a))$ and $(b,f(b))$. At start it looks like nothing, but we can come to the conclusion that as the point $b$ approaches the point $a$, the secant line gets closer to the tangent line of the point $(a,f(a))$. Look the graph:

In that way, we can use the idea of Limits to understand that while $b$ approaches $a$, the slope of the line that pass through the points $(a,f(a))$ and $(b,f(b))$ approaches the slope of the tangent line of the point $(a,f(a))$.

With this understanding, we can now draw a right triangle to decipher some values and finally get to the equation that defines the tangent of the point $(a,f(a))$:

So we have a right triangle in which the adjacent side is equals to $b-a$ and the opposite side is $f(b)-f(a)$. With this information we can now apply the slope equation:

$$m=\frac{f(b)-f(a)}{b-a}$$

But we do not care about the slope of $b$ in any point, we want $b$ to be closer and closer of $a$. For this we can just apply the concept of limits:

$$m=\underset{b\to a}{\lim }\frac{f(b)-f(a)}{b-a}$$

Here then is the derivative. The derivative is nothing more than the slope of a tangent line. And to represent it we do not use $m$, instead we use the following:

$$f^{\prime }(a)=\underset{b\to a}{\lim }\frac{f(b)-f(a)}{b-a}$$

Another common way to represent it is introducing another variable called $h$, which represents $b-a$:

$$f^{\prime }(a)=\underset{b\to a}{\lim }\frac{f(a+h)-f(a)}{h}$$