1. Write the precise definition for a function f(x) defined for all values of x to be a differentiable function.
The derivative of a function $\displaystyle f$ is a function defined as, all the values in the domain such that,
$\displaystyle \lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$ exist (and at least one exists).
The map of the function is defined as its limit value.