Derivative of Square Function

We are to find the derivative of a function without a given point. Therefore, we need to implement the slope formula in an abstract manner. You may know that the slope formula can be expressed as $\displaystyle{\lim_{\Delta x \to 0}{f(\Delta x) \over {\Delta x}} = {{f(x\ +\ \Delta x)\ -\ f(x)}\over{\Delta x}}}$ .

Plugging in our function, we get: $\displaystyle{\lim_{\Delta x \to 0}{{\Delta f(x)}\over{\Delta x}}=\lim_{\Delta x \to 0}{{x^{2} + (\Delta x)^{2} + 2x\Delta x - x^{2}}\over{\Delta x}} = \lim_{\Delta x \to 0}{{(\Delta x)^{2} + 2x\Delta x}\over{\Delta x}} = \lim_{\Delta x \to 0}{\Delta x + 2x} = \boxed{2x}}$ .

We are asked to provide ${f'(x)}\over{x}$ , so ${2x\over{x}} = \boxed{2}$ .