Derivatives of Products and Quotients Michael Freeze MAT 151 UNC - - PowerPoint PPT Presentation

Derivatives of Products and Quotients

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

1 / 13

Product Rule

Let f (x) = u(x) · v(x) where both u′(x) and v ′(x)

• exist. Then

f ′(x) = u(x) · v ′(x) + v(x) · u′(x). That is, the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative

• f the first.

2 / 13

Product Rule Verification

Product Rule d dx [u(x) · v(x)] = u(x) · v ′(x) + v(x) · u′(x) Let f (x) = (3x2 + 2)(2x − 1). (a) Express f (x) as a polynomial in standard form and find f ′(x) by differentiating term-by-term. (b) Use the product rule to find f ′(x) directly.

3 / 13

Product Rule Examples

Product Rule d dx [u(x) · v(x)] = u(x) · v ′(x) + v(x) · u′(x) Use the product rule to find the derivative of the function.

f (x) = (x − 5)(1 − 2x)

4 / 13

Product Rule Examples

Product Rule d dx [u(x) · v(x)] = u(x) · v ′(x) + v(x) · u′(x) Use the product rule to find the derivative of the function.

y = (x5 − 2x3)

• x − 1

x

• 5 / 13

Product Rule Examples

Product Rule d dx [u(x) · v(x)] = u(x) · v ′(x) + v(x) · u′(x) Use the product rule to find the derivative of the function.

g(x) = (x + 1)(√x + 2)

6 / 13

Quotient Rule

Let f (x) = u(x)

v(x) where both u′(x) and v ′(x) exist

and v(x) = 0. Then f ′(x) = v(x) · u′(x) − u(x) · v ′(x) [v(x)]2 . That is, the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

7 / 13

Quotient Rule Verification

Quotient Rule d dx u(x) v(x)

• = v(x) · u′(x) − u(x) · v ′(x)

[v(x)]2 Let f (x) = 3x2−3x+2 x . (a) Express f (x) as a sum of power functions and find f ′(x) by differentiating term-by-term. (b) Use the quotient rule to find f ′(x) directly.

8 / 13

Quotient Rule Examples

Quotient Rule d dx u(x) v(x)

• = v(x) · u′(x) − u(x) · v ′(x)

[v(x)]2 Use the quotient rule to find the derivative of the function.

f (x) = 2x−5

3x+1

9 / 13

Quotient Rule Examples

Quotient Rule d dx u(x) v(x)

• = v(x) · u′(x) − u(x) · v ′(x)

[v(x)]2 Use the quotient rule to find the derivative of the function.

y = x2−2x+4

x−1

10 / 13

Equation of Tangent Line

Find an equation of the tangent line to the curve

f (x) = x x − 2 at x = 3.

11 / 13

Average Cost

The total cost (in hundreds of dollars) to produce x units of perfume is C(x) = 3x + 2 x + 4 . Find the average cost for each production level. (a) 10 units (b) 20 units (c) x units (d) Find the marginal average cost function.

12 / 13

Memory Retention

Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f (t) = 90t 99t − 90. Find the rate at which the number of facts remembered is changing after the following numbers

• f hours.

(a) 1 (b) 10

13 / 13