Introduction to :

The "derivative of f at a," written f ' (a), is a number that is equal to the slope of the function f at a.

For any differentiable function f there is another function, known as the derivative of f and written (x). We write '(x) to show that this is a function.

We can calculate the derivative function using the limit definition in the same way we calculated the value of the derivative at a point using the limit definition.

Practice:

Example 1

Let f(x) = x2. Calculate f'(x).


Example 2

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


Example 3

Let f(x) = x3. Find f'(x).


Example 4

Let f(x) = x2. We found that f'(x) = 2x. Find f(3).


Exercise 1

Let f(x) = x3. Use the formula we found for f'(x) to evaluate

  •  f'(4)

Exercise 2

Let f(x) = x3. Use the formula we found for f'(x) to evaluate

  • f'(-5)

Exercise 3

Let f(x) = x3. Use the formula we found for f'(x) to evaluate


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