# 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*.