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 f ' (x). We write f '(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.
Let f(x) = x2. Calculate f ' (x).
Let f be a line. That is, f(x) = mx + b where m and b are constants. What is f ' (x)?
Let f(x) = x3. Find f ' (x).
Let f(x) = x2. Given that f ' (x) = 2x, what is f ' (3)?