## Differentiation by First Principles Formula Proof

Differentiation is the process by which we can get from a function, $$f(x)$$, to its gradient function, $$f'(x)$$, i.e. a function that outputs the gradient of the curve at a point.

Suppose we have some function, $$y=f(x)$$ :

(*graph)

Lets now take two points on the graph. The first will just be a generic point, $$(x,f(x))$$. For the second point we will use $$(x+\delta x, f(x+ \delta x))$$, where $$\delta x$$ means "a small change in x", i.e. our new point is our old point plus a small change:

(*graph)

If we join the two points with a line segment, we can see that the gradient of this line segment (we'll call it $$m$$ for now) is given by:

$m = \frac{f(x+\delta x)-f(x)}{\delta x}$

from the usual formula that $$gradient=\frac{\Delta y}{\Delta x}$$.

Now we can see that if we make $$\delta x$$ smaller, the line segment joining the two points comes closer to being the gradient of the curve at the point $$(x, f(x))$$.  I.e., as $$\delta x \to 0$$,  $$m \to f'(x)$$:

(*animation)

Therefore, writing this in mathematical notation, we get:

$f'(x)=\lim_{\delta x \to 0} \frac{f(x+\delta x)-f(x)}{\delta x}$

where $$lim$$ means the 'limit' (i.e. what it tends towards) as $$\delta x$$ tends to zero. We have now derived the formula for the gradient function. Applying this formula is referred to as 'differentiating from first principles'.