13.2 Intuitive Idea of Derivatives

We have all encountered derivatives at one point or another. In the simplest way, we can say it is all about the slope (It is the slope of the tangent line to the graph of a function). Although this definition is not technically incorrect, they do not capture the essence of the derivative and the brilliance of the mathematicians who discovered it.

In mathematics, we use mathematical functions to model how the behavior of a physical or even non-physical quantity depends on another quantity. For example, consider the function that defines Gravitational Force between two masses as shown below.

The function \(\mathrm{F}\) calculates the gravitational force between two masses at a distance \(r\). We say that the force, “F”, is the dependent variable while ” \(r\) “, the distance, is the independent variable. The values that ” \(r\) ” takes on determine the values of “F”. The gravitational force is a function of distance and we write \(\mathrm{F}=\) \(\mathrm{F}(\mathrm{r})\).

A function is a quantity that depends on another quantity. The exact nature of the dependence is described by the formula of the function. What is not described by the formula, however, is the sensitivity of the dependence. Consider the function \(\mathrm{F}=\mathrm{F}(\mathrm{r})\) that we mentioned before. Although we do have the formula for the gravitational force we do not know how the force responds to changes of the independent variable \(\mathrm{r}\), the distance. That is if we change the distance by a small amount by how much will the force change? This is the question that gave birth to the concept of the derivative.

The derivative of a function \(f=f(x)\) tells us how rapidly the function \(f(x)\) varies when we change the argument \(x\) by a tiny amount. If ” \(d f\) ” represents a tiny change of the function \(f(x)\) and ” \(d x\) ” represents a tiny change of the independent variable ” \(x\) ” then the derivative is the proportionality factor.

Derivative of \(f(x)=\frac{d f}{d x}\)