3.2 The Power RuleThurs Oct 22

Do Now

Find the derivative of:

Quiz Review

• Retakes by Tues

Computations of Derivatives

• Thm- For any constant c,

• Note, when y = c, the slope of that line is always horizontal. Therefore, its derivative must equal 0

• Thm- Let f(x) = x, then

• Proof:

• Note: This means that the derivative of any linear function is equal to the coefficient

Power Rule• Let’s take a look at the different powers of x.

Can you see the pattern in the table?

F(x) F’(x)

1 0

X^1 1

X^2 2x

X^3 3x^2

X^4 4x^3

Power Rule cont’d

Power Rule - For any real number n,

Note: The power rule works for negative exponents, as well as fraction exponents.

Ex 3.1Find the derivatives of

Ex 3.2Find the derivatives of

General Derivative Rules

• Thm- If f(x) and g(x) are differentiable at x and c is any constant, then

• 1)

• 2)

• 3)

General Deriv. Rules• Remember, to rewrite any expressions so

they have exponents! And split the expression into separate terms!

• You try: Find the derivative of each:• 1)

• 2)

• 3)

Closure

• Hand in: Find the derivative of:

• 1)

• 2)

• HW: p.139 #7-13 odds, 21-27odds, 33, 49

3.2 Power Rule cont’dFri Oct 23

• Do Now:

• Find the derivative of:

• 1)

• 2)

HW Review p.139 #7-13 21-27, 33, 49

• 7) -32 33) 32s - 24• 9) 1/3 49) a) • 11) b)• 13) c)• 21)• 23)• 25)• 27)

Derivative of e^x

• The derivative of f(x) = e^x is

Finding tangent line equations

• Now that we have the power rule, it is much easier to find the equations of tangent lines.

• EX: Find the tangent line to the graph at x = 2

Differentiability and Continuity

• If f is differentiable at x = c (the derivative is defined at c) then f is also continuous at c

Derivative Info

• The derivative can tell us when a function is increasing (+), decreasing (-), or horizontal (0)

• This makes finding the vertex of a function easier

Closure

• Journal Entry: How does the power rule work? What does a derivative tell us about a function?

• HW: p. 139 #15-19 all, 29 31 43 45

3.2 Power Rule PracticeFri Oct 26

Do Now

Find the derivative of the following:

1)

2)

HW Review: 139 #15-19, 29 31 43 45

• 15) y = 32x - 48 45) A = f(x)• 16) y = -2/125 x + 3/25 B = h(x)• 17) y = -3x - 32 C = g(x)• 18) y = 1/12 x + 4/3• 19)a) 12e^x b) 25-8e^t c) e^(t-3)• 29) 0• 31) • 43) A - 3, B - 1, C - 2, D - 3

Derivative Rules so far:

• Power Rule• E^x• Constant Rule• Breaking up functions into individual terms

Practice

• Green book worksheet p.184 #5-23

Closure

• Journal Entry: How does the graph of a function relate to the graph of its derivative?

• HW: Finish worksheet p.184 #5-23

HW Review p.184 5-23 odds

• 5) 3x^2 -2 21)• 7) 6x 23)• 9) 0• 11) • 13)• 15) -5x^(-3/2) -2• 17) • 19)

HW Review p.184 5-23 evens

• 6)• 8)• 10) -4• 12) • 14)• 16)• 18)• 20)• 22)