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

0cdxd

Examples of the Constant Rule

Function Derivative y = 34 dy/dx = 0 y = 2 y’ = 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

)(' 1 nnxxf

Examples of the Power Rule

Function Derivative

23)(' xxf

3)( xxg 32

32

31

3

131

)('x

xxdxd

xg

2

1)(

xxh 3

32 22

xxx

dxd

dxdh

3)( xxf

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.

4)1(4)1('

4)(')(3

34

f

xxfxxf

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.

)()( xfdxdcxfc

dxd

Using the Constant Multiple Rule

Function Derivative

x2 2

211 21222

xxx

dxd

xdxd

dxdy

54 2t ttt

dtd

tdtd

tf 58

542

542

54 )2()('

3 22

1

x

35

35

32

3

132

21

21

xxx

dxd

dxdy

Using Parentheses when Differentiating

This is the same as the Constant Multiple Rule, but it can look a lot more organized!

Examples:Original Rewrite Differentiate Simplify

325x

325 xy 4

25 3' xy 42

15'

xy

2)3(7

x263xy )2(63' xy xy 126'

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:

)(')(')()( xgxfxgxfdxd

)(')(')()( xgxfxgxfdxd

Using the Sum and Difference Rules

Function Derivative

54)( 3 xxxf 43)(' 2 xxf

xxx

xg 232

)( 34

292)(' 23 xxxg

The Derivatives of Sine and Cosine Functions

Make sure you memorize these!!!

xxdxd cossin

xxdxd sincos

Using Derivatives of Sines and Cosines

Function Derivative

xsin2 xcos2

2sin x

2cos x

xx cos xsin1

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

002

21)( stvgtts

Congratulations!!!

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