Sec 2 Basic Differentiation Rules Finding the DerivativeRate

Sec 2: Basic Differentiation Rules Finding the Derivative/Rate of Change

I. The Constant Rule � The derivative of a constant is 0. That is, if c is a real number, then 1. Sketch a graph to demonstrate this rule. 2. Prove that the derivative of a constant, c, is 0.

Ex 1: Find the derivatives A. y=7 B. f(x) = 0 B. f’(x) = 0 C. s(t) = -3 C. s’(t) = 0 D. y = kπ² D. y’ = 0 A. y’ = 0

II. The Power Rule � If n is a rational number, then the function is differentiable and

Ex 2: Find the Derivative A. f(x) = x³ f’(x) = 3 x² B. g(x) = g’(x) = C. y= y’ = -2/x³

III. The Constant Multiple Rule � If f is differentiable and c is a real number, then c·f(x) is also differentiable and � Prove the Constant Multiple Rule.

Ex 3: Find the derivative � f(x) � g(x) �y = 2 x² = -4 x³ = (¾)x³

Ex 4: Find the derivatives FUNCTION REWRITE DERIVATIVE SIMPLIFY

HOMEWORK � Pg 113 #1 -9 odds, 25– 30 odds

Write the following rules: � Constant � Power Rule � Constant Multiple Rule

IV. Sum and Difference Rule � The sum or difference of two differentiable functions is differentiable and is the sum or difference of their derivatives. � Prove the Sum Rule

Ex 4: Find the derivative � f(x) � g(x) = x³ - 4 x + 5 =

Ex 5: Equation of a Tangent Line � Find the equation of the line tangent to the graph f(x) = x² when x = -2.

HOMEWORK � Pg 113 #11 -18, 31 -34, 53, 55

AP Practice 1. No Calculator 2. 3.

VI. Derivatives of Sine & Cosine � Find the derivative. 1. y = 2 sin x 2. y= 3. f(x) = x + cos x

V. Average Velocity POSITION → VELOCITY � Position Function – A function, s(t), that gives the position of an object as a function of time. � So � Therefore,

Ex 1: Average Velocity � If a billiard ball is dropped from a height of 100 ft, then its height s at time t is given by the position function, s(t) = -16 t² + 100 where s is in feet and t is in seconds. Find the average velocity for the following time intervals. A. [1, 2] B. [1, 1. 5] C. [1, 1. 1]

HOMEWORK � PG 115 #21 -24, 81 -86, 87 -90 *Just find avg. velocity on 87 -90

AP Practice

Using Derivative to find Velocity � The position of any free falling object can be given by the position function Where is initial velocity, and g is gravity. � Velocity is initial height, at a particular instant is

� Velocity � Speed can be negative, 0, or positive. is the ___________ of velocity.

Ex 4: Velocity � At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t) = -16 t² + 16 t + 32 where s is in feet and t is in seconds. A. When does the diver hit the water? B. What is the diver’s velocity at impact? C. How fast was the diver going?

AP Practice

HOMEWORK � Pg 115 #87 -90 *find instantaneous velocity at each endpoint and compare to previous night’s answers , 91 -94