Definition

Let f(x) be our function. Then

$$f'(c) = \displaystyle\lim_{x \to c} \displaystyle\frac{f(x) - f(c)}{x-c}$$

provided the limit exists.

Essentially,

$$\displaystyle\frac{f(x) - f(c)}{x-c}$$

tell us the average rate of change between x and c. As $x \to c$ the average rate of change more closely approximates the instantaneous rate of change at c. This limiting process allows us to find the slope of the function at any point.

Another useful way to express the derivative is the following

$$f'(x) = \displaystyle\lim_{\bigtriangleup x \to 0} \displaystyle\frac{f(x + \bigtriangleup x) - f(x)}{\bigtriangleup x}$$

Compare this definition with the previous one to ensure they are the same.

Notice that wherever our function attains a local maximum or minimum the tangent line will be horizontal. Because the slope of any horizontal line is zero, wherever there is a max or min for our function the derivative will be equal zero. Therefore, one way to find a maximum or minimum value is to take the derivative and set it equal to zero and solve for x.

For example, if the function f(x) = -x² + 100x told us the area of a rectangle with sides whose perimeter is 200, how could we determine the rectangle in this class with the greatest area? For now, assume that we know the derivative of this function is f'(x) = -2x + 100. Then

$$\begin{eqnarray*}f'(x) & = & -2x + 100\\ 0 &=& -2x + 100\\ 2x &=& 100\\ x &=& 50\\ \end{eqnarray*}$$

Therefore, a rectangle which best maximizes area in this case is a square with all sides length equal to 50.


How do we determine the derivative of a function?