Notation for Derivatives¶

We will use derivatives a lot, so we introduce notation for derivatives.

Lagrange's notation¶

We write the derivative of a function $f$ at $a$ as $$ f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}. $$ This is called Lagrange's notation. So, to write the result of the last example on the previous page, we can let $f(x) = x^2$ and then say $f'(x) = 2x$.

With this notation, you can think of $f'$ itself as a thing; it's a function that takes in a number, and outputs the derivative of $f$ at that number. Often the function $f'$ is called the derivative of $f$.

Leibniz's notation¶

We can also write $$ \frac{d}{dx} x^2 = 2x. $$ This is called Leibniz's notation, and it can be handy because we didn't need to define a function just to talk about the derivative of $x^2$. So, $\frac{d}{dx}$ sees whatever you put in front of it as $\text{function}(x)$, and then gives you $\text{function}'(x)$; that is, $$ \frac{d}{dx} f(x) = f'(x). $$

The $\frac{d}{dx} x^2$ is actually a shorthand for $\frac{d(x^2)}{dx}$, with $x^2$ on the top like that. Originally $d(x^2)$ was an infinitely small difference of $x^2$ values, given by $$ d(x^2) = (x + \text{infinitely small number})^2 - x^2, $$ and similarly, $dx$ was an infinitely small difference of $x$ values. Then $\frac{d(x^2)}{dx}$ was just two infinitely small numbers divided by each other. Nowadays we have limits instead of infinitely small numbers, and the derivative is more than division; it's a limit of whatever the division gives. The notation is still being used though.