Derivative

The derivative of a function \(f(x)\) is the limit, if any, of the expression
\[ \frac{f(x+\delta)-f(x)}{\delta} \]
as \(\delta\) tends to zero.

The value of the derivative at \(y\) is typically witten as one of
\[ \frac{\mathrm{d}f}{\mathrm{d}x}\bigg|_y \;=\; \frac{\mathrm{d}f}{\mathrm{d}x}(y) \;=\; \overset{\cdot}{f}(y) \;=\; f^\prime(y) \]
The derivative represents the rate of change of the function with reprect to its argument or, equivalently if we plot the function as a graph its tangent, or slope.

By repeating the process \(n\) times we recover higher order derivatives, typically written as one of
\[ \frac{\mathrm{d}^nf}{\mathrm{d}x^n}\bigg|_y \;=\; \frac{\mathrm{d}^nf}{\mathrm{d}x^n}(y) \;=\; f^{(n)}(y) \]