We'll use the limit definition, putting in x instead of a:

Therefore f ' (x) = 2x. If we plug in 1 for x we find f ' (1) = 2, which agrees with our earlier calculation.

With formulas, the limit definition of the derivative function is

This is the same as the limit definition of the derivative at a point, but with x instead of a. When we evaluate the derivative of f at a point, we take a value and plug it in for x in the definition above.

Example 2

Let f be a line. That is, f(x) = mx + b where m and b are constants. What is f ' (x)?

Let's see if we can make an educated guess before we dive into the limit.

Since derivative means "slope" and lines have a constant slope, we should expect f ' (x) to just be the slope of the line. Let's see if our suspicions are confirmed. Here we go:

We already thought of m as the "slope" of a line mx + b. It's nice that when we calculated the derivative we got m, since the derivative is also supposed to be the "slope." This means that for any line

f(x) = mx + b, the derivative function is a constant function:

f ' (x) = m for every value of x.

Example 3

Let f(x) = x^{3}. Find f ' (x).

We use the limit definition of the derivative:

But we'll leave it to you to check that

(x + h)^{3} = x^{3} + 3xh^{2} + 3x^{2}h + h^{3}.

Therefore

Once we have a formula for the derivative of a function, we can calculate the value of the derivative anywhere.

Example 4

Let f(x) = x^{2}. Given that f ' (x) = 2x, what is f ' (3)?

Since f ' (x) = 2x, we have

Just plug the number into the function. That's all there is to it.