33 Rules of Differentiation

Objective Students will be able tohellip

bull Apply the Power Rule Sum and Difference Rule Quotient and Product Rule for differentiation

bull Find higher order derivatives

What is the slope of y=2 y=-5 y=10

What is frsquo(x) if f(x)=2

DERIVATIVE OF A CONSTANT PROOF

Let f(x) =c where c is a constant

If c is a constant then

00

lim

)()(lim)(

0

0

hh

cch

xfhxfxf

h

h

0)( cdx

d

EXPLORATION

Using the definition of the derivative find the derivative of each function below

1) f(x) = x1

2) f(x)=x2

3) f(x)=x3

Do you notice anything about f(x) and frsquo(x)

The Power Rule

If n is a rational number then the function f(x)=xn is differentiable and

Examples Find the derivative

a) f(x) = x4

b) g(x)=

c) y=

1 nn nxxdx

d

3 x

2

1

x

The Constant Multiple Rule

If f is a differentiable function and c is a real number then cf is also a differentiable function and

Examples

)()( xfcxcfdx

d

2

2

3

5)()4

2)3

5

3)()2

4)()1

xth

xy

xxg

xxf

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

What is the slope of y=2 y=-5 y=10

What is frsquo(x) if f(x)=2

DERIVATIVE OF A CONSTANT PROOF

Let f(x) =c where c is a constant

If c is a constant then

00

lim

)()(lim)(

0

0

hh

cch

xfhxfxf

h

h

0)( cdx

d

EXPLORATION

Using the definition of the derivative find the derivative of each function below

1) f(x) = x1

2) f(x)=x2

3) f(x)=x3

Do you notice anything about f(x) and frsquo(x)

The Power Rule

If n is a rational number then the function f(x)=xn is differentiable and

Examples Find the derivative

a) f(x) = x4

b) g(x)=

c) y=

1 nn nxxdx

d

3 x

2

1

x

The Constant Multiple Rule

If f is a differentiable function and c is a real number then cf is also a differentiable function and

Examples

)()( xfcxcfdx

d

2

2

3

5)()4

2)3

5

3)()2

4)()1

xth

xy

xxg

xxf

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

EXPLORATION

Using the definition of the derivative find the derivative of each function below

1) f(x) = x1

2) f(x)=x2

3) f(x)=x3

Do you notice anything about f(x) and frsquo(x)

The Power Rule

If n is a rational number then the function f(x)=xn is differentiable and

Examples Find the derivative

a) f(x) = x4

b) g(x)=

c) y=

1 nn nxxdx

d

3 x

2

1

x

The Constant Multiple Rule

If f is a differentiable function and c is a real number then cf is also a differentiable function and

Examples

)()( xfcxcfdx

d

2

2

3

5)()4

2)3

5

3)()2

4)()1

xth

xy

xxg

xxf

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

The Power Rule

If n is a rational number then the function f(x)=xn is differentiable and

Examples Find the derivative

a) f(x) = x4

b) g(x)=

c) y=

1 nn nxxdx

d

3 x

2

1

x

The Constant Multiple Rule

If f is a differentiable function and c is a real number then cf is also a differentiable function and

Examples

)()( xfcxcfdx

d

2

2

3

5)()4

2)3

5

3)()2

4)()1

xth

xy

xxg

xxf

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

The Constant Multiple Rule

If f is a differentiable function and c is a real number then cf is also a differentiable function and

Examples

)()( xfcxcfdx

d

2

2

3

5)()4

2)3

5

3)()2

4)()1

xth

xy

xxg

xxf

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

The Sum and Difference Rules

If f and g are differentiable functions then their sum (f +g) and their difference (f-g) are also differentiable and

)()()()( xgxfxgxfdx

d

h

xgxfhxghxfxgxf

dx

dh

)()()()(lim)()(

0

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

EXAMPLES

234

234

3

232

)()3

73410)()2

54)()1

xxx

xh

xxxxxg

xxxf

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

Does y=3x4-x2+1 have any horizontal tangent lines

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

THE PRODUCT RULE YEAH

If f and g are differentiable functions then their product fg is differentiable and

)()()()( xfxgxgxffgdx

d

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

EXAMPLES

)12()()3

113)()2

4523)()1

2

2

2

rrrh

xxxxg

ttttf

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

THE QUOTIENT RULE WOOHOO

If f and g are differentiable functions then their quotient fg gne0 is differentiable and

2)()()()()(

)(

)(

xg

xgxfxfxg

xg

xf

dx

d

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

EXAMPLES

15

52)3

1

2)()2

1

25)1

2

2

2

2

xx

xxy

xx

xxh

x

xy

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

Rewriting Before Differentiating

2

2

)52()()3

)3)(2()()2

6

3)()1

xxg

xxxg

xxxf

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

x

xxp

xx

xxxh

3

5)()5

158

127)()4

2

2

2

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

Find the equation of the line tangent to the curve at point (1-1) x

xy

2

42 2

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

Second and Higher Order Derivatives

yrsquo = dydx is the first derivative with respect to x

Second Derivative

Third Derivative

Fourth Derivative

Fifth Derivative5

5)5(

4

4)4(

3

3

2

2

dx

ydy

dx

ydy

dx

ydy

dx

ydy

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples

Examples

1 Find yrsquorsquo given y=3x2+5

2 Find y(5) given y=x4+2x3-3x2+10x+1

• 33 Rules of Differentiation
• What is the slope of y=2 y=-5 y=10
• EXPLORATION
• The Power Rule
• The Constant Multiple Rule
• The Sum and Difference Rules
• EXAMPLES
• Does y=3x4-x2+1 have any horizontal tangent lines
• THE PRODUCT RULE YEAH
• EXAMPLES
• THE QUOTIENT RULE WOOHOO
• EXAMPLES (2)
• Rewriting Before Differentiating
• Slide 14
• Find the equation of the line tangent to the curve at poin
• Second and Higher Order Derivatives
• Examples