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Derivatives

Derivatives

At a Glance - The Derivative Function

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.

When using the limit definition, instead of using f(a), we just use f(x). Since x can be any value, the resulting limit will be a new brand new function. If we plug a point a into this function, the output will be f ' (a), the derivative of f at a.

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. What is f ' (x)?


Example 3

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


Example 4

Let f(x) = x2. Given that f ' (x) = 2x, what is 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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