Warm-Up

Find a and b such that f(x) is continuous on the entire real line.

3,2

31,

1,2

x

xbax

x

xf

Calculus Warm-Up

• For a function to be differentiable at x=c, what conditions must be met?

1) f must be continuous at c

2) The derivative to the left and

right of c must be equal

Calculus HWQ

• Is this function continuous as x = 2?

• Is the function differentiable at x = 2?

yes

no

2.2 Basic Differentiation Rules 2015

Colorado National MonumentGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Recall from last lesson: To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner cusp

vertical tangent discontinuity

f x x 2

3f x x

3f x x 1, 0

1, 0

xf x

x

Continuous & DifferentiableContinuous but NOT differentiable everywhere:

**Remember: differentiablility implies continuity but continuity does NOT imply differentiability**

Continuous AND differentiable everywhere :

Discontinuous

You try…

At point c, decide whether these graphs are:

a. Continuous

b. Differentiable

c. Both

d. Neither

e)

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

0dc

dx

example: 3y

0y

The derivative of a constant is zero.

If we find derivatives with the difference quotient:

2 22

0limh

x h xdx

dx h

2 2 2

0

2limh

x xh h x

h

2x

3 33

0limh

x h xdx

dx h

3 2 2 3 3

0

3 3limh

x x h xh h x

h

23x

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

(Pascal’s Triangle)

2

4dx

dx

4 3 2 2 3 4 4

0

4 6 4limh

x x h x h xh h x

h

34x

2 3

We observe a pattern: 2x 23x 34x 45x 56x …

1n ndx nx

dx

examples:

4f x x

34f x x

8y x

78y x

power rule

We observe a pattern: 2x 23x 34x 45x 56x …

d ducu c

dx dx

examples:

1n ndcx cnx

dx

constant multiple rule:

57dx

dx

When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

4 47 5 35x x

(Each term is treated separately)

d ducu c

dx dx

constant multiple rule:

sum and difference rules:

d du dvu v

dx dx dx d du dv

u vdx dx dx

4 12y x x 34 12y x

4 22 2y x x

34 4dy

x xdx

Example:Find the horizontal tangents of: 4 22 2y x x

34 4dy

x xdx

Horizontal tangents occur when slope = zero.34 4 0x x

3 0x x

2 1 0x x

1 1 0x x x

0, 1, 1x

Plugging the x values into the original equation, we get:

2, 1, 1y y y

(The function is even, so we only get two horizontal tangents.)

4 22 2y x x

4 22 2y x x

2y

4 22 2y x x

2y

1y

4 22 2y x x

4 22 2y x x

First derivative (slope) is zero at:

0, 1, 1x

34 4dy

x xdx

Function Rewrite Differentiate Simplify

3

5

2y

x 35

2y x

415'

2y x

4

15'

2y

x

3

5

2y

x 35

8y x

415'

8y x

4

15'

8y

x

2

7

3y

x 27

3y x

14'

3y x

2

7

3y

x 263y x ' 126y x

Often it is helpful to rewrite the function before differentiating:

Function Derivative

3 4 5f x x x 2' 3 4f x x

4

33 22

xg x x x 3 2' 2 9 2g x x x

Practice:

Function Derivative

2f x x 1'f x

x

23 6 5g x x x x 2' 36 45g x x x

Practice:

2

0

2

Consider the function siny

We could make a graph of the slope: slope

1

0

1

0

1Now we connect the dots!

The resulting curve is a cosine curve.

sin cosd

x xdx

2

0

2

We can do the same thing for cosy slope

0

1

0

1

0The resulting curve is a sine curve that has been reflected about the x-axis.

cos sind

x xdx

Derivatives of Sine and Cosine Functio

sin cos

ns

cos sind d

x x x xdx dx

Function Derivative

2sinf x x ' 2cosf x xsin

2

xy

1' cos

2y x

Practice:

cosy x x ' 1 siny x

Higher Order Derivatives:

dyy

dx is the first derivative of y with respect to x.

2

2

dy d dy d yy

dx dx dx dx

is the second derivative.

(y double prime)

dyy

dx

is the third derivative.

4 dy y

dx is the fourth derivative.

We will learn later what these higher order derivatives are used for.

Example:

4 3 23 2 6 1f x x x x x

f x

f x

f x

4f x

5f x

3 212 6 12 1x x x

236 12 12x x

72 12x

72

0

Homework:

Pg.115 Larson Calculus 8th ed. : 31-63 odd, 75, 89

http://college.cengage.com/mathematics/blackboard/content/larson/calc8e/calc8e_solution_main.html?CH=07&SECT=a&TYPE=se