Today in Calculus• Go over homework• Derivatives by limit definition• Power rule and constant rules for derivatives• Homework

When f′(a) fails to exist

Where the tangent does not exist:

points of discontinuity (function does not exist)

at corners

at cusps

Also at vertical tangents (slope is undefined)

Theorems• If f has a derivative at x = a, then f is continuous at

x = a.• Intermediate Value Theorem for derivatives: If a & b

are any two points in an interval on which f is differentiable, then f ′ takes on every value between f ′(a) & f ′(b).

Note: A function can be continuous but not differentiable but if it is differentiable it has to be continuous

Finding the Derivative

Finding the Derivative

Finding the Derivative

Derivative Rules

Power rule:

f(x) f′(x)

3x

7x

x2

x3

1n ndx nx

dx

Derivative Rules

Sum and difference rules: If u and v are differentiable functions of x, then

f(x)= 3x2+5x – 12

d du dvu v

dx dx dx

Examples

Examples