The derivative of a function

In the previous section we have derived formulas for calculating the derivative of various functions. For example, if the function is

f(x)=x^3

we can use the formula

a_u = 3u^2

to calculate the slope of the tangent at point $A$ on the graph, where $u$ is the $x$ -coordinate of $A$ . We can regard this formula as a function, where $u$ is the input (to the machine). But let us use $x$ for the input, as usual. So we have the function

\begin{array}{cll} x & \\ \downarrow &\\ \boxed{f^\prime} & \text{rule: } 3x^2\\ \downarrow &\\ y & \\ \end{array}

This function, $f^\prime(x)=3x^2$ , which we have derived from the function $f(x)=x^3$ is fittingly called the derivative of $f$ , and is denoted by $f^\prime$ (say "f prime of x").

For every function $f$ can we get its derivative $f^\prime$ . So far we had to do this the hard way, using the differential quotient. But we will see later that finding the derivative $f^\prime$ of a function $f$ can be done by applying some rules, the so called rules of differentiation.

Example 1

With this new notation we can say that the derivative of $f(x)=x^3$ is

f^\prime(x)=3x^2

and this rule calculates the slope of the tangent to the graph of $f$ at $x$ . For example, the slope of the tangent to the graph of $f(x)=x^3$ at $x=2$ (or point $A(2 | 8)$ ) is

f^\prime(2)=3\cdot 2^2=12

Let us summarise ...

Definition 1

Consider a function $f$ . The derivative of $f$ , denoted by

f^\prime

is the function which calculates the slope of the tangent to the graph of $f$ . Thus, if $A$ is a point on the graph with $x$ -coordinate $x$ , then

Equation 1

f^\prime(x) = \text{slope of tangent at $A(x\vert f(x))$}

Definition of a function

or using the differential quotient

Equation 2

f^\prime(x) = \lim_{x\rightarrow 0}\frac{f(x+h)-f(x)}{h}

The process of finding $f^\prime$ is called differentiation, and we say that we differentiate $f$ to find its derivative $f^\prime$ :

\begin{array}{c} f \\ \downarrow &\\ f^\prime & \\ \end{array}