12-02: Derivatives

chain rule

The ________ is an effective way of differentiating a composite function by first: differentiating the outer function with respect to the inner function, then multiplying by the derivative of the function.

Distance

________ and direction that an object has moved from an origin over a period of time.

Velocity

________: a vector quantity with both magnitude and direction.

IROC

a derivative

To differentiate radicals

express it as a power with a rational exponent

To differentiate a power of x that is in the denominator

express as a power with a negative exponent

Use y=mx+b, plug in (x,y) from tangent point and slope

isolate for b

Composite function

one function being subbed into another function

The chain rule is an effective way of differentiating a composite function by first

differentiating the outer function with respect to the inner function, then multiplying by the derivative of the function

Second derivative

f(x) determined by differentiating the first derivative (same process, just starting from the first derivative)

Speed

a quantity or scale that describes the magnitude of motion but does not describe direction

Velocity

a vector quantity with both magnitude and direction

Answer for velocity questions can be either + or

and the sign indicates the direction at which the object is travelling

when v(t) = 0

object is at rest

when v(t) > 0

object is moving in a + direction

when a(t) > 0

object is accelerating; velocity is increasing

when a(t) < 0

object is decelerating; velocity is decreasing

if v(t) a(t) > 0

object is speeding up

if v(t) a(t) > 0

object is slowing down

Second derivative

f’’(x) determined by differentiating the first derivative (same process, just starting from the first derivative)

Possible speed and acceleration values

when v(t) = 0: object is at rest

when v(t) > 0: object is moving in a + direction

when v(t) < 0: object is moving in a - direction

when a(t) > 0: object is accelerating; velocity is increasing

when a(t) < 0: object is decelerating; velocity is decreasing

if v(t) • a(t) > 0: object is speeding up

if v(t) • a(t) > 0: object is slowing down