Derivatives

differentiability and continuity

if f(x) is differentiable, it MUST be continuous

differentiability

f(x) is differentiable at f(s) if and only if…

• it is continuous at f(s)

• f’(s) exists

  • f’(s) exists if lim h→s exists; so

• lim h→ 0 f(exists

  • then find the limit of

    

  • then find the limit of

    

  • if lim h→ s^- = lim h→s⁺ , then lim h→s exists, and f(x) is differentiable at f(s)

key examples of functions where f’(x) is NOT differentiable

continuity

f(x) is continuous at f(s) if and only if…

• f(s) exists; and

• lim x→s exists; and

• lim x→s = f(s)

derivatives

definition of derivative is…

then add derivative rules from 2/24 lec

notes about exam:

• 50 minutes

• everything from today (2/24) and thursday (2/26) will be on the exam

• six questions

• multiple choice

• each point is about 16.6 points

• points are removed for every wrong answer, but less than 16.6 points

• there will be a practice exam uploaded today

• scientific calculator allowed but not graphing calculator

• can bring my own scratch paper

limit definition:

lim x→a f(x)-f(a)/x-a

• use this if x is going to a specific point

derivative rules

take notes from lec 2/24; section 2.4

how to compute derivatives

power functions

(xⁿ)’ = nx^n-1

• also true for fractional and negative powers

example: (x² + 3x^5)’ = (x²)’ + (3x⁵)’ → 2x + 3 × 5x⁴ → 2x + 15x⁴

General rules:

• f(x) ± g(x))’ = f’(x) ± g’(x)

• (sin x)’ = cos x

• (cos x)’ = -sin x

notes from lec 2/26

computing derivatives in an efficient way (not using the definition of derivative) cont’d from lec 2/24

section 2.4 i believe

Product rule

(f(x) g(x))’ = f’(x) _^x g(x) + f(x) _^x g’(x)

Quotient rule

(f(x)/g(x))’ = (f’(x) _^x g(x) - f(x) _^x g’(x))/g(x)²

• (lo _x d’hi - hi _x d’lo)/ lo _x lo → lodihi - hidilo/ lolo