By: Hunter FaulkAnthony StephensMeghan Troutman

• Defines the rate of change in a given function.

Lim f(x+h) – f(x) h→0 h

Example: f(x)= f’(x)=2x http://images.google.com/

• Newton and Leibniz shared credit for the development of the differential and integral calculus. They both contributed in the math world by developing new areas of calculus such as differentiation, integration, curves, and optics. Both published their facts that included integral and derivative notation we still use today.

The Derivative of the Functions Will Use Notation That Depends on the

FunctionFunction First Derivative Second Derivative

F(x) F’(x) F’’(x)

G(x) G’(x) G’’(x)

Y Y’ or dy dx

Y’’ or

• Definition of the Power Rule The Power Rule of Derivatives gives the following: For any real number n,the derivative of f(x) = xn is f ’(x) = nxn-1 which can also be written as

• Definition of the quotient ruleRule for finding the derivative of a quotient of two functions. If

both f and g are differentiable, then so is the quotient f(x)/g(x). In abbreviated notation, it says

(f/g) = (′ gf − ′ fg )/′ g2.

• Definition of the product rule Rule for finding the derivative of a product of

two functions. If both f and g are differentiable, then (fg) = ′ fg + ′ f′g.

• Chain rule The formula is . Another form of

the chain rule is .

• Rule number 1 If y= , then

ExampleIf y= , then

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Number oney=

Number two y=

Work Page

Answers to Power Rule Problems

• Problem One

• Problem Two

Original Function is blue. Derivative is red.

• Used when given composite functions. A composite function is a function inside another function. F(G(x))

• First you take the derivative of the outside function, while leaving the inside function alone. Then you multiply this by the derivative of the inside function, with respect to its variable x. If y= f(g(x)), then y’=

Chain Rule Examples

• Problem one

• Problem two

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Work Page

Answer to Chain Rule Problem One

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Problem one

Answer to Chain Rule Problem Two

Problem two

• Bottom function times the derivative of the top minus the top function times the derivative of the bottom. Then divide the whole thing by the bottom function squared. If f(x)= , then f’(x)=

Quotient Rule Examples

• Problem one

• Problem two

Work Page

• Problem one

Answer to Quotient Problem Number Two

• Problem Two

(x-2 + x-6)(-3x-4 + 8x-9)– (x-3- x-8)(-2x-3- 6x-7)(x-2 + x-6) 2

When a function involves two terms multiplied together, we use the Product Rule.

To find the derivative of two things multiplied by each other, you multiply the first function by the derivative of the second, and add that to the second function multiplied by the derivative of the first.

If F(x) = uv, then f’(x) = u

Product Rule

Exampleswww.first90days.wordpress.com

• Problem One

f(x)= (9x2+4x)(x3-5x2)

• Problem Two (x5-11x8)

Work Page

F’(x)= (9x2+4x)(3x2-10x) + (x3-5x2)(18x+4)

Answer to Product Rule Problem Two

Y’= (5x4-88x7) + (x5 – 11x8)

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• Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable while treating the other variables as unspecified functions of x.

• taking d of another variable

• Y2 -> 2y

Examples ofImplicit Differentiation

• Problem OneX2 + y2 = 5

• Problem TwoX2 + 3xy + y2 = ∏

Work Page

Solving Implicit Differentiation Problem

One2x =2y = 0

2x = -2y =

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Answer to Implicit Differentiation Problem

Two

2x + 3x + 3y + 2y = 0 (3x +2y) = -2x - 3y

= -2x – 3y 3X +2Y

A method for finding the derivative of functions such as y = xsin x and

Y= lny=xln3 =x(0)+ ln3(1) =ln3 = ln3

Examples of Logarithmic Differentiation

• Problem one y=

• Problem two y= http://images.google.com/

Work Page

Y=lny=xlnx

=x( )+lnx

= (1+lnx)

Answers to Logarithmic Differentiation Problem

TwoY=

=sinx + lnxcosx= ( +lnx(cosx))

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1978 AB 2 Let f(x) = (1 - x)2 for all real numbers x, and let

g(x) = ln(x) for all x > 0. Let h(x) = ( 1 - ln(x))2. a. Determine whether h(x) is the composition

f(g(x)) or the composition g(f(x)). b. Find h (x). ′c. Find h″(x). d. On the axes provided, sketch the graph of h.

FRQ Answer

A) f(g(x)) = f(lnx) = (1-lnx)2g(f(x)) = g((1-x)2) = ln((1-x)2)

Therefore h(x) = f(g(x))B) h’(x) = 2(1-lnx)(1/x) = 2 lnx-1

C) h’’(x) = 2 x(1/x) – (lnx-1) x2

= 2 * 2 – lnx x2D) The Inflection point is at (-e2, 1)

The minimum is at (e, 0 )

FRQ Example

1977 AB 7 BC 6 Let f be the real-valued function defined by f(x)

= sin3(x) + sin3|x|. a. Find f (x) for x > 0. ′b. Find f’(x) for x < 0. c. Determine whether f(x) is continuous at x =

0. Justify your answer. d. Determine whether the derivative of f(x)

exists at x = 0. Justify your answer.

FRQ Answer

A) For x > 0F(x) = sin3x + sin3x = 2sin3xF’(x) = 6sin2xcosx

B) For x > 0f(x) = sin3x + sin3(-x) = sin3x - sin3x = 0F’(x) = 0

C) f(0) = 0Lim f(x) = lim 2sin3x = 0Lim f(x) = lim 0 = 0

Since Lim f(x) = 0 = f(0), the function of f is continous at x = 0D) F’(x) = lim f(x+h) – f(x) if the limit exists hAt x = 0Lim f(h) - f(0) = 0 hTherefore, lim f(h) - f(0) and so f’(0) exists and equals 0. h

X ->0+X ->0+

X ->0- X ->0+

X ->0+

H -> 0

H -> 0

H -> 0

Bibliography• http://www.mathwords.com/f/formula.htm• http://images.google.com/imgres?imgurl=http://www.francis.edu/

uploadedImages/Math/blackboard_math.gif&imgreful• www.open.salon.com• www.babble.com • www. school-clipart.com• www.first90days.wordpress.com• www.schools.sd68.bc.ca • http://numericalmethods.eng.usf.edu/anecdotes/newton.html• http://gardenofpraise.com/ibdnewt.htm• http://scienceworld.wolfram.com/biography/leibniz/html• http://calculusthemusical.com/wp-content/uploads/2008/02/matheatre-power-

rule.mp3©Copyright. Hunter Faulk. Megan Troutman. Anthony Stevens. February 19, 2010