4-34-3Derivative of a Quotient Derivative of a Quotient of Two Functionsof Two Functions

Pages 136- 141Pages 136- 141

Katie Warmbold Katie Warmbold

Jillian Defrietas Jillian Defrietas

• This chapter allows you to solve This chapter allows you to solve “catching up problems” in which one “catching up problems” in which one moving object wants to reach another moving object wants to reach another moving object. moving object.

• Property: Derivative of a Quotient of Two Functions:– If y = u/v, where u and v are differentiable

functions, and v doesn’t equal 0, then

u’v – uv’ y’ = v^2

Words – Derivative of top times bottom, minus top times derivative of bottom, all divided by bottom squared

• In order to differentiate, or find the derivative, In order to differentiate, or find the derivative, some functions, its important to remember the some functions, its important to remember the Property: Derivatives of Sine and Cosine Property: Derivatives of Sine and Cosine FunctionsFunctions..– d/dx(sinx) = cosXd/dx(sinx) = cosX– d/dx(cosx) = -sinXd/dx(cosx) = -sinX• The The chain rulechain rule, explained in 3-7, is applied , explained in 3-7, is applied

in numerous problems when finding the in numerous problems when finding the derivative.derivative.

• Differentiate: Differentiate: (5x – 2)^7(5x – 2)^7

y = (4x +9)^3y = (4x +9)^3• Differentiate:Differentiate:

d d ( ( 5 5 ) ) ddx ( 7x^3)x ( 7x^3)

• Differentiate:Differentiate: sin5xsin5x

f(x) = 8x - 3f(x) = 8x - 3

Detailed ProcedureDetailed Procedure

• Problem # 1:Problem # 1: (5x – 2)(5x – 2)77

y= (4x +9)y= (4x +9)33

Use the Use the PropertyProperty to solve problem to solve problem(HINT(HINT:: Derivative Derivative oof topf top timestimes bottombottom, , minusminus toptop

timestimes derivative of bottomderivative of bottom, all , all divided bydivided by bottom bottom squaredsquared.).)

The chain rule, as in 4-2, also is applied when The chain rule, as in 4-2, also is applied when finding derivativesfinding derivatives

y’ = y’ = 7(5x-2)^6(5)7(5x-2)^6(5) * * (4x+9)^3(4x+9)^3 - - (5x - 2)^7(5x - 2)^7 * * 3(4x + 9)^2(4)3(4x + 9)^2(4)(4x + 9)^6(4x + 9)^6

= = (5x - 2)^6 (80x + 339)(5x - 2)^6 (80x + 339) (4x + 9)^6(4x + 9)^6

• Differentiate:Differentiate: sin5xsin5x f(x) = 8x - 3f(x) = 8x - 3 1.)1.) cos5x(5)(8x - 3) - sin5x(8) cos5x(5)(8x - 3) - sin5x(8) f’x = (8x - 3)^2f’x = (8x - 3)^2 2.)2.) 5cos5x(8x - 3) - 8sin5x5cos5x(8x - 3) - 8sin5x f’x = (8x - 3)^2f’x = (8x - 3)^2

• Black Hole ProblemBlack Hole Problem : : Ann Astronaut’s spaceship gets trapped in the Ann Astronaut’s spaceship gets trapped in the

gravitational field of a black hole! Her velocity, v(t) gravitational field of a black hole! Her velocity, v(t) miles per hour is given:miles per hour is given:

10001000 v(t) = 3 - tv(t) = 3 - t a.)a.) How fast is she going when t=1? When t = 2? When How fast is she going when t=1? When t = 2? When

t = 3?t = 3? B.)B.) Write an equation for the acceleration function, Write an equation for the acceleration function,

a(t).a(t).

C.)C.) What is her acceleration when t=1? When t = 2? What is her acceleration when t=1? When t = 2?

When t = 3?When t = 3?

• A.)A.) a(1) = 250 mph/hr a(1) = 250 mph/hr a(2) = 1000 mph/hra(2) = 1000 mph/hr a(3) = 1000/0 mph/hr = no answera(3) = 1000/0 mph/hr = no answer• B.) B.) 10001000 a(t) = (3 - t)^2a(t) = (3 - t)^2• C.)C.) a(1) = 250 mph/hr a(1) = 250 mph/hr

a(2) = 1000 mph/hra(2) = 1000 mph/hr a(3) = 1000/0 mph/hr = no answera(3) = 1000/0 mph/hr = no answer

• This section taught you how to find an This section taught you how to find an equation for the derivative function in equation for the derivative function in one step when the function contains a one step when the function contains a quotient of two other functions.quotient of two other functions.

• The The Property; Derivative of a Quotient of Property; Derivative of a Quotient of Two FunctionsTwo Functions was discussed and was discussed and applied.applied.