What are derivatives really?

TL;DR: Derivatives measure the slope at a given point.

When we have a linear function like:

$$f(x)=2x$$

We know the slope is 2 because as x changes by 1, y changes by 2. So the derivative of 2x is 2.

But if we have a quadratic function (or any other function where the slope isn’t constant) like $x^2$ then the slope changes at different points. We can approximate it at a given point by changing x slightly and then calculating the slope. The smaller the change the more exact our slope is:

$$\frac{df}{dx} = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

That’s the definition for derivatives with limits. A limit describes the value the function will reach as it gets closer to a value in this case 0. Now from the limit definition we can derive the derivative of $x^2$:

$$\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h \xrightarrow{h \to 0} 2x$$

which generalizes to the power rule:

$$\frac{d}{dx} x^n = n \cdot x^{n-1}$$

There are multiple notations for derivatives. The Leibniz one used here is: $\frac{df}{dx}$. Then there’s also the Lagrange notation which uses apostrophes so $f'$ for the first derivative. Newton uses a dot and it specifically means the derivative with respect to time $\dot{f}$, you’ll see it mostly in physics.

Now here’s a list of common functions and their derivatives:

$f(x)$	$f'(x)$	Notes
$c$ (a constant)	$0$	A flat line has no slope
$x$	$1$	Power rule with $n=1$
$x^n$	$n \cdot x^{n-1}$	Works for any real $n$, not just integers
$e^x$	$e^x$	Function is the same as the derivative
$\ln(x)$	$\frac{1}{x}$	Only defined for $x > 0$
$\sin(x)$	$\cos(x)$	Differentiate four times and you’re back to $\sin(x)$
$\cos(x)$	$-\sin(x)$	Watch the minus sign
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$	Power rule with $n=\frac{1}{2}$

Now knowing what derivatives are, there are a few nice side effects that are helpful for analyzing functions:

1. Critical points are where the slope of a function equals 0
2. The Second Derivative Test (SDT) tells us what kind of critical point that is.

So for example $x^2$ has exactly 1 Critical Point at $(0,0)$ because the second derivative is 2, which is positive, so it’s a local minimum. So we can use derivatives to maximize a function and more generally find its critical points and classify them using SDT.

In this post I just scratched the surface of calculus, there is way more to it. If you want to go deeper I’d recommend Dr. Chris Tisdell’s videos. He has a great playlist called Engineering Mathematics