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 FunctionDerivative • y = 34 dy/dx = 0 • y = 2y’ = 0 • s(t)= -3 s’(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) = xn is differentiable and
Examples of the Power Rule FunctionDerivative
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 x4 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: • s0 = initial position • v0 = initial velocity • g = acceleration due to gravity (-32 ft/sec2 or -9.8 m/sec2) • Example: Find the velocity at 2 seconds of an object with position s(t) = -16t2 + 20t + 32. • 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 7th Edition!!