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 \(x_0.\)

\[\frac{df(x_0)}{dx} := \lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}\]

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

\[\frac{df(x)}{dx} := \lim_{x_0 \rightarrow 0} \frac{f(x+x_0)-f(x)}{x_0}.\]

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