2.2 Basic Differentiation Rules and Rates of Change

Objective: Find the derivative using the Constant Rule, Power Rule, Constant Multiple Rule, and Sum and Difference Rules. Find the derivatives of the sine

function and of the cosine function.

Ms. BattagliaAB/BC Calculus

Thm 2.2 The Constant RuleThe derivative of a constant function is0. That is, if c is a real number, then

Examples:Function Derivativea.y = 7 dy/dx =b.f(x) = 0 f’(x) =c.s(t) = -3 s’(t) =d.y = kπ2, k is constant y’ =

Thm 2.3 The Power RuleIf n is a rational number, then the function f(x) = xn is differentiable and

d/dx[xn]=nxn-1

For f to be differentiable at x=0, n must be a number such that xn-1 is defined on an interval containing 0.

Using the Power Rulea. b. c.

Finding the Slope of a Graph

Find the slope of the graph of f(x) = x4 whena. x = -1 b. x = 0 c. x = 1

Finding an Equation of a Tangent LineFind an equation of the tangent line to the graph of f(x) = x2 when x = -2

Thm 2.4 The Constant Multiple RuleIf f is a differentiable function and c is a real number, then cf is also differentiable and

d/dx [cf(x)] = cf’(x)

Thm 2.5 The Sum and Difference RulesThe sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f+g (or f-g) is the sum (or difference) of the derivatives of f and g.

Using the Constant Multiple Rule

a. b. c.

Using the Constant Multiple Ruled. e.

Original Function

Rewrite Differentiate Simplify

Using Parenthesis When Differentiating

Using the Sum and Difference Rules

a. f(x) = x3 – 4x + 5 b.

Derivatives of the Sine & Cosine Functions

Theorem 2.6

Derivatives Involving Sines & Cosinesa. b. c.

the average velocity is

Rates of Change

The position function for a projectile is s(t) = –16t2 + v0t + h0, where v0  represents the initial velocity of the object and h0 represents the initial height of the object.

An object is dropped from the second-highest floor of the Sears Tower, 1542 feet off of the ground. (The top floor was unavailable, occupied by crews taping for the new ABC special "Behind the Final Behind the Rose Final Special, the Most Dramatic Behind the Special Behind the Rose Ever.")

(a) Construct the position and velocity equations for the object in terms of t, where t represents the number of seconds that have elapsed since the object was released.

(b) Calculate the average velocity of the object over the interval t = 2 and t = 3 seconds.

(c) Compute the velocity of the object 1, 2, and 3 seconds after it is released.

(d) How many seconds does it take the object to hit the ground? Report your answer accurate to the thousandths place.

(e) If the object were to hit a six-foot-tall man squarely on the top of the head as he (unluckily) passed beneath, how fast would the object be moving at the moment of impact? Report your answer accurate to the thousandths place.

AB: Read 2.2, Page 115 #3-30, 31-57 odd

BC: Read 2.2, Page 115 #3-30, 31-57 odd, 97-100

Classwork/Homework