# Derivatives

Let \(X \subseteq \mathbb{R}\) be a subset of real numbers, \(x_0 \in X\) and \(f: X \rightarrow \mathbb{R}\) be a function that maps elements of \(X\) to real numbers. The function \(f\) is **differentiable** at \(x_0\) if the limit of \(\frac{f(x)-f(x_0)}{x-x_0}\) as \(x \rightarrow x_0\) converges to a real number \(\frac{df(x_0)}{dx}\) which is the **derivative** at \(x_0.\)

If \(f\) is differentiable \(\forall \, x_0 \in X\) then \(f\) is differentiable on \(X.\) If the limit does not exist then \(\frac{df(x_0)}{dx}\) is undefined and \(f\) is not differentiable at \(x_0.\)

Let \(X \subseteq \mathbb{R}\) be a subset of real numbers, \(x_0 \in X\) and \(f: X \rightarrow \mathbb{R}\) be a function that maps elements of \(X\) to real numbers. The **derivative** of the function \(f\) is

Differentiability and continuity

Higher-order derivatives

Derivatives of sums

Derivatives of powers

Product rule

Quotient rule

Chain rule

Local maxima and minima

L’Hôpital’s rule

Taylor expansion