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# 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:

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