Linearization

The process of taking a function and ignoring the terms above delta x.

The definition of a gradient is Rise over Run

$\frac{Rise}{Run} = Gradient$

This can be rearranged to:

$Run \times Gradient = Rise$

or

$(x - p) \times f'(p) = Rise$

Given the Taylor series formula

$\large g(x) = \sum\limits_{n=0}^{\infty} \frac{f^{(n)}(p)}{n!}(x-p)^n$

We get

$\large g_1(x) = f(p) + f'(p)(x-p)$

If we express it using p

$\large g_1(p + \Delta p) = f(p) + f'(p)(\Delta p)$

then

$\large g_1(x + \Delta p) = f(x) + f'(x)(\Delta x)$

Finally, we get

$\large f(x + \Delta x) = \sum\limits_{n=0}^{\infty}\frac{f^{(n)}(x)}{n!}\Delta x^n$

which can be succinctly rewritten as:

$\large f'(x) = \frac{f(x + \Delta x) - f(x)}{\Delta x} + O(\Delta x)$