• Introduction
• Iteration
  • Summation
  • Product
  • Set Notation
  • Nested Iteration
  • Conditions on indices
  • Union and Intersection
  • General Iteration
• General Symbols
  • Cases
  • Function Composition
  • Set Containment
  • Kronecker
• Operations
  • Factorial
  • Binomial
• Differentiation
  • Derivative
  • Partial Derivative
  • Gradient

Derivative $\operatorname{f}'(x)$, $\frac{d \operatorname{f}}{d x}$

The derivative measures how much a value changes with respect to another. For a function $\operatorname{f}$, we can ask how much the value $\operatorname{f}(x)$ changes, if we change $x$ a tiny amount.

The notations $\operatorname{f}'(x)$ ("f prime of x") and $\frac{d \operatorname{f}}{d x}$ ("df [over] dx") mean the same thing. The first one is often used in basic 1D calculus, the second is probably seen more often higher dimensional variants and physics. The "d" is a "differential" quantity, more or less meaning "something very very small".

There are a few different notations, but mainly we have:

$$\frac{d \operatorname{f}}{d x} = \lim\limits_{h \rightarrow 0} \frac{\operatorname{f}(x + h) - \operatorname{f}(x)}{h}$$

The $\lim$ is a so called limit and $h\rightarrow 0$ means "let h approach 0". When using a computer, we can't do that (we can with symbolic and automatic differentiation, but not with the simple method) actually compute this. Instead, we approximate this by using a small value of $h$. How do we choose the value of h? Well, that can be more complicated, so usually you will see some slightly hacky fixed value, that is "very" small, like $10^{-7}$. As stated before, this is a very simple code that works, but could be improved in many ways, if needed.

So in math notation, that is called the "difference quotient":

$$\frac{d \operatorname{f}}{d x} \approx \frac{\operatorname{f}(x + h) - \operatorname{f}(x)}{h}$$

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