FindingADerivative

derivative

Finding a derivative!
=====================
The definition of an derivative is as follows: 

$$
\lim_{\Delta x \to 0} { {f(x+\Delta x) - f(x)} \over \Delta x}
$$

Given that: 

$$
y = f(x) = x^2
$$

Let's find the derivative: 

$$
{\lim_{\Delta x \to 0} { {f(x+\Delta x) - f(x)} \over \Delta x} } = {\lim_{\Delta x \to 0} { {\left(x+\Delta x\right)^2 - x^2} \over \Delta x} }
$$
$$
{\lim_{\Delta x \to 0} { {\left(x^2+2x\Delta x + \Delta x\right) - x^2} \over \Delta x} }
$$
$$
{\lim_{\Delta x \to 0} { {x^2- x^2+2x\Delta x + \Delta x } \over \Delta x} }
$$
$$
{\lim_{\Delta x \to 0} { {2x\Delta x + \Delta x } \over \Delta x} }
$$
$$
\therefore f'\left( x \right) = 2x
$$