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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.