Section 2-2: Basic Differentiation Rules and Rates of Change Eun Jin Choi, Victoria Jaques, Mark Anthony Russ

Brief Overview The Constant Rule Power Rule Constant Multiple Rule Sum and Difference Rules Derivatives of Sine and Cosine Functions How to find Rates of Change (Velocity and Acceleration)

The Constant Rule The derivative of a constant function is 0. That is, if c is a real number, then

Examples of the Constant Rule Function Derivative y = 34dy/dx = 0 y = 2y’ = 0 s(t)= -3s’(t) = 0 Notice the different notations for derivatives. You get the idea!!!

The Power Rule If n is a rational number, then the function f(x) = x n is differentiable and

Examples of the Power Rule Function Derivative

Finding the Slope at a Point In order to do this, you must first take the derivative of the equation. Then, plug in the point that is given at x. Example: Find the slope of the graph of x 4 at -1.

The Constant Multiple Rule If f is a differentiable function and c is a real number, then cf is also differentiable and So, pretty much for this rule, if the function has a constant in front of the variable, you just have to factor it out and then differentiate the function.

Using the Constant Multiple Rule FunctionDerivative

Using Parentheses when Differentiating This is the same as the Constant Multiple Rule, but it can look a lot more organized! Examples: OriginalRewriteDifferentiateSimplify

The Sum and Difference Rules The sum (or difference) of two differentiable functions is differentiable. The derivative of the sum of two functions is the sum of their derivatives. Sum (Difference) Rule:

Using the Sum and Difference Rules FunctionDerivative

The Derivatives of Sine and Cosine Functions Make sure you memorize these!!!

Using Derivatives of Sines and Cosines FunctionDerivative

Rates of Change Applications involving rates of change include population growth rates, production rates, water flow rates, velocity, and acceleration. Velocity = distance / time Average Velocity = ∆distance / ∆time Acceleration = velocity / time Average Acceleration = ∆velocity / ∆time

Rates of Change (con’t) In a nutshell, when you are given a function expressing the position (distance) of an object, to find the velocity you must take the derivative of the position function and then plug in the point you are trying to find. Likewise, if you are trying to find the acceleration, you must take the derivative of the velocity function and then plug in the point you are trying to find.

Using the Derivative to Find Velocity Usual position function: – s 0 = initial position – v 0 = initial velocity – g = acceleration due to gravity (-32 ft/sec 2 or -9.8 m/sec 2 ) Example: Find the velocity at 2 seconds of an object with position s(t) = -16t t – First take the derivative: s’(t) = -32t + 20 – Then, plug in 2 to find the answer: s’(2) = -44 ft/sec

Congratulations!!! You have now mastered Section 2 of Chapter 2 in your very fine Calculus Book: Calculus of a Single Variable 7 th Edition!!