Derivative Definition: f'(a) = lim(h->0) [f(a+h)-f(a)]/h

One-Sided Derivatives:

• f'+(a) = lim(h->0+) [f(a+h)-f(a)]/h
• f'-(a) = lim(h->0-) [f(a+h)-f(a)]/h
• f differentiable iff f'+(a) = f'-(a) (finite)

Hierarchy: Differentiable => Continuous (not vice versa)

Absolute Value:

• |f(x)| non-differentiable at simple zeros of f (where f'!=0)
• |f(x)| differentiable at x=a if f(a)!=0 or if f(a)=0 and f'(a)=0

Max/Min:

• max(f,g) = $\frac{f+g+|f-g|}{2}$
• min(f,g) = $\frac{f+g-|f-g|}{2}$
• Non-differentiable where f=g and f'!=g'

Rolle's Theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) => f'(c)=0 for some c in (a,b)

LMVT: f continuous on [a,b], differentiable on (a,b) => f'(c)=[f(b)-f(a)]/(b-a) for some c in (a,b)

IVT: f continuous on [a,b], k between f(a) and f(b) => f(c)=k for some c in (a,b)

Darboux: If f' exists on [a,b], then f' has the intermediate value property (no jumps)

Inverse Function: (f^(-1))'(y) = 1/f'(x) where y = f(x), provided f'(x)!=0

Key Functions:

• $x^n$ sin$\frac{1}{x}$, f(0)=0: differentiable at 0 iff n>=2
• [x]: discontinuous at integers, f'=0 between integers
• {x}: discontinuous at integers, f'=1 between integers