guide to differentiation - a start on calculus

Hey, welcome aboard calculus. I’ll make this interesting, I promise.

figure: Consider $f(x)$. What is the rate of change between these points?

You can see, for a function $f(x)$, the average rate of change from one x value to another is easily calculated by rise over run, as $\frac{f(c+\Delta x)-f(c)}{\Delta x}$

figure: Consider a lower value of h.

Okay, that wasn’t much different. We still calculate the average rate of change the same way.

Combining this simple logic with the idea of limits, we can calculate the instantaneous rate of change (the slope) of a point.

The instantaneous rate of change at the point $(x,f(x))$ can be calculated with $\underset{\Delta x\to 0}{\lim }\frac{f(c+h)-f(c)}{h}$

We found the slope for a point on the function. We show this as $f^{\prime }(x=c)$, meaning the slope of $f(x)$ at the point where $x=c$.

As a general thing, $f^{\prime }(x)$ is a function that can give you the slope for any x value you have.

We can find that, $f^{\prime }(x=c)=\underset{\Delta x\to 0}{\lim }\frac{f(c+\Delta x)-f(c)}{\Delta x}$.

and finally,

$f^{\prime }(x)=\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}$.

Remember this and leave it inside your head as the First Principles.

A wonderful explanation of First Principles is here.

This, my friends, is the start of calculus.