Basic Differentiation Rules and Rates of Change The Constant Rule The derivative of a constant function is 0. For any real number, c

y The slope of a horizontal line is 0.

x The derivative of a constant function is 0.

1 PROOF: Let f(x) = c, Then by the limit definition of the derivative,

EX. #1 Using the Constant Rule

Function Derivative

2 The Power Rule The derivative of the term axn , where a and n are real numbers, is

STEPS:

1. Multiply the coefficient by the variable's exponent. If no coefficient is stated – in other words, the coefficient equals 1– the exponent becomes the new coefficient. 2. Subtract 1 from the exponent.

EX.#2: Use the power rule to find f '(x) if:

3 Finding the Slope of a Graph

EX.#3: Find the slope of the graph of f(x) when:

The slope of a graph at a point is the value of the derivative at that point.

a. x = 2

b. x = 0

c. x = ­2

4 Finding an Equation of a Tangent Line

EX.#4: Find the equation of the tangent line to the graph of f(x) when:

The derivative of f is:

a. At x = 2, y ' = 3

b. At x = 0, y ' = ­1

c. At x = ­2 , y ' = ­5

5 Derivatives of the Sine and Cosine Functions

EX #5: Derivatives Involving Sines and Cosines

Function Derivative

6 RATES OF CHANGE

The derivative can determine slope and can also be used to determine the rate of change of one variable with respect to another.

It is customary to describe the motion of an object moving in a straight line with either a horizontal or vertical line, from some designated origin, to represent the line of motion.

• Movement right or upward is considered positive direction. • Movement left or downward is considered negative direction.

Vocabulary rate of change derivative instantaneous rate of change } displacement distance traveled

7 FORMULAS TO RECOGNIZE

General Position Function

g ­ gravitational constant v0 ­ initial velocity s0 ­ initial height Free Fall Constants English Metric – 32 ft/sec2 – 9.8 m/sec2

Velocity Function

1. Tells how fast an object is moving 2. Tells the direction of motion object moves forward ⇒positive object moves backward ⇒ negative 3. Velocity is derivative of position function with respect to time.

Average Velocity of an Object over time Interval

Distance Rate = Time

Change in distance V = avg Change in time

8 The average velocity between P t1 and t2 is the slope of the secant line, and the instantaneous velocity at t1 is the slope of the tangent line.

t1 = 1 t2

9 Speed

1. absolute value of velocity 2. measures the rate of motion, regardless of direction 3. non­negative velocity

Speed =

Acceleration

1. rate at which velocity changes 2. measures how quickly the body picks up or loses speed. 3. derivative of velocity or second derivative of position function.

10 EX #6: A coin is dropped from the top of a building that is 1362 feet tall.

A. Write the position function.

B. Write the velocity function.

C. Find the average velocity on [1, 2].

Vavg =

11 D. Find instantaneous velocities when

t = 1

t = 2

E. When will coin reach ground?

F. What is the velocity at impact?

G. Convert (f) to miles per hour.

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