# Derivative Rules

IROC: a derivative

In essence, the steps to differentiating (from above)

Power Rule

1. Drop the exponent’s value down to the front of the term

2. Subtract 1 from the exponent

1. If there is a coefficient, then multiply the dropped value by the coefficient

2. If you have an exponent of 0, the term is 1 (anything to the power of 0 is 1)

• 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

Determining equations from equations

1. Find tangent point (if given x value use that to find y by subbing into the original equation)

2. Find the derivative

3. Sub x value into the derivative to find the slope of the tangent

4. Use y=mx+b, plug in (x,y) from tangent point and slope - isolate for b

# The Product Rule

Instead of expanding out, there is another way to find the derivative of p(x)=f(x)g(x)

p’(x) = f’(x)g(x) + g’(x)f(x)

# The Quotient Rule

Method used to differentiate functions in the form q(x) = f(x)/g(x)

# The Chain Rule

Composite function: one function being subbed into another function

• What’s on the inside of the brackets/radical/etc

• Parent function (what’s on the outside)

The chain rule:

f’(x) = g’(h(x)) • h’(x)

• A composite function g(h(x)) consists of an outer function g(x) and an inner function h(x). 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

# Velocity, Acceleration, and Second Derivatives

## Second Derivatives

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

On a graph:

Slope of f(x)

Derivative

+

is +, above x axis

0

is an x intercept

-

is -, below x axis

Slope of f’(x)

Derivative

+

is +, above x axis

0

is an x intercept

-

is -, below x axis

## Using Derivatives to Analyze the Motion of Objects Travelling in a Straight Line

Displacement

Velocity

Acceleration

Definition

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

rate of change of displacement of an object with respect to time

rate of change of velocity with respect to time

Relationship

s(t)

s’(t)

s”(t)

Possible Units

m

m/s

m/s²

Speed and velocity are different

• 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

Possible velocity v(t) and acceleration a(t) 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