Introduction
Derivatives are functions that are (as you can assume) derived from other functions. Derivatives tell you the slope (a.k.a. rate of change) of a function at a given point.For example, assume we have a function $f(x)=\frac{x^3}{3}$:
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The derivative of this function is $f'(x)=x^2$. Notice the $'$ after the $f$. This is one of the ways you can denote a derivative. It's read 'f prime of x'.
We can graph the derivative on top of $f(x)$ to get an idea of what is going on:
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If we compute $f'(1)$ we see get the result of $1$. This means that the slope at $f(1)=1$. You can easily tell if the slope is $1$ if it appears to make a $45^\circ$ turn from either positive axis. Likewise, $f'(1)=-1$, therefore, the slope at $f(-1)$ is -1.
You can compute $f'(x)$ for any $x$ in the domain of $f$.
Definition
The simplest definition of a derivative involves computing a limit.
Imagine that we want to get the tangent line of the curve $f(x)=\frac{x^3}{3}$ at $x=1$. We could start by making an approximation by drawing a line through two points on the curve. those points are at a distance $h$ on the x-axis.
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If we would infinitely keep making $h$ smaller, we would eventually get the tangent line at the point $x=1$. If we can get the slope of the tangent line (which measures the instantaneous rate of change at a point), we have the derivative. This is the limit definition of the derivative:
$$\lim_{h\to0}=\frac{f(x+h)-f(x)}{h}$$
Example:
Find the derivative of $f(x)=4x^2$.
Start with the limit definition of the derivative.
$f'(x)=\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h}$
Substitute.
$f'(x)=\lim\limits_{h\to0}\frac{4(x+h)^2-4x^2}{h}$
Expand numerator.
$f'(x)=\lim\limits_{h\to0}\frac{4x^2+8xh+4h^2-4x^2}{h}$
Subtract.
$f'(x)=\lim\limits_{h\to0}\frac{8xh+4h^2}{h}$
Divide.
$f'(x)=\lim\limits_{h\to0} (8x+4h)$
Solve the limit.
$f'(x)=8x$
The limit definition of the derivative is one of the building blocks for some shortcuts for finding derivatives more quickly. I will write more posts concerning derivatives soon. I hope this was useful. All feedback is welcome!