REFERENCE SHEET

Derivatives & Taylor expansion

The derivative is a local linear model; Taylor expansion adds controlled higher-order terms.

Before this Limits & continuity

f(a+h) = f(a) + f′(a)h + o(h)

Definitions

Derivative

f′(a)=limₕ→₀ [f(a+h)−f(a)]/h when the limit exists.

Critical point

An interior point at which the derivative is zero or does not exist; these are candidates for local extrema.

Taylor polynomial

The degree-n polynomial at a with coefficients f⁽ᵏ⁾(a)/k!, for 0≤k≤n.

Key theorems

Mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), then some c∈(a,b) satisfies f′(c)=[f(b)−f(a)]/(b−a).

Taylor remainder

If f is Cⁿ⁺¹ on an interval containing a and x, then f(x)=Σₖ₌₀ⁿ f⁽ᵏ⁾(a)(x−a)ᵏ/k! + f⁽ⁿ⁺¹⁾(ξ)(x−a)ⁿ⁺¹/(n+1)! for some ξ between a and x.

Chain rule

If g is differentiable at x and f at g(x), then (f∘g)′(x)=f′(g(x))g′(x).

Notable applications

Optimization

Derivatives identify stationary candidates and monotonicity intervals.

Numerical approximation

Taylor bounds quantify approximation error, rather than just providing a polynomial guess.

Work it out

Try before revealing

01 / Compute

Differentiate exp(x²).

Hint

Apply the chain rule.

Solution

The derivative is 2x exp(x²).

02 / Bound

Bound the error of 1+x+x²/2 as an approximation to exp(x) on |x|≤0.1.

Hint

Bound the third derivative on [−0.1,0.1].