Introduction to Derivatives

Now that we have seen several ways of thinking about the derivative and have actually performed one calculation, we would like to develop some handy techniques to compute derivatives. In a way, this is like learning the multiplication tables after you understand what multiplication means.

In this unit, we will learn how to compute the derivatives of rational functions. You may remember that polynomials are functions which are built in the simplest way possible---namely, through addition and multiplication. Rational functions are simply functions obtained by taking the quotient of two polynomials. For example, $\frac{x}{x^2 + 1}$ is a rational function. Later, we will learn how to differentiate more complicated expressions.

Notation

We have seen the notation $y^\prime(x)$ to denote the derivative of the function $ y $ at the point $ x
$ . It is also convenient to use the notation
$\frac{dy}{dx} (x)$
to denote the same quantity. This is natural since it recalls the origin of the derivative as a rate of change.

Before we get started though, we need to consider an important application of the derivative in approximating functions.