Differentiation - Quotient Rule

©S w2k0b1Q3v tKWuFtWai TSJonf4tawdakrZet 5LnLsCG.j 4 vA6lOlZ 9rZi7gHhvtUsK FreecshearQvneUdL.p s NMRaYdAer MwciAtAhN 2I9nCfWianjiatSey 6CRaJlLcluTlJuAst.g Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____

Date________________

Differentiation - Quotient Rule

Differentiate each function with respect to

x.

1) y = 2

2x⁴ − 5 2) f

(

x

)

^{= 2}

x⁵ − 5

3) f

(

x

)

^{= 5}

4x³ + 4 4) y =

4x³ − 3x²

4x⁵ − 4

5) y =

3x⁴ + 2

3x³ − 2 6) y =

4x⁵ + 2x²

3x⁴ + 5

7) y =

4x⁵ + x² + 4

5x² − 2 8) y =

3x⁴ + 5x³ − 5 2x⁴ − 4

-1-

©R B2n0w1s3s PKnuytYaJ fShogfRtOwGadrTen hLyLHCB.4 s tA1lFlU 1rviOgZhJthse Tryersae6rHvzeIdJ.h W AMkaPdCeQ MwtiGtFhB dIenjfuienAietfeT XCvaTl1cwuylUuKsY.3 Worksheet by Kuta Software LLC

9) y =

x³ − x² − 3

x⁵ + 3

10) y =

x⁴ + 6 3 − 4

x⁻⁴

11) y =

4x⁴ − 4x² + 5

2

x

5

3 + 3

Critical thinking question:

12) A classmate claims that

(

g^f

)

^{' =} g^f'^' for any functions f and g. Show an example that proves your classmate wrong.

-2-

©7 f2V021V3O nKMuJtCaF VSYoSfgtfwFaGrmeL 8LpLCCP.M Q mAFl7lL orxiqgDh0tpss LrFezsyeIrrvReNds.P Q uMSa0d4eL twii7t6hz YI0nsfMionEiMtzeL ECia7lDctu9lfuesU.f Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____

Date________________

Differentiation - Quotient Rule

Differentiate each function with respect to

x.

1) y = 2

2x⁴ − 5 dy

dx

= − 2 ⋅ 8x³

(

²x⁴ − 5

)

²

= − 16x³ 4x⁸ − 20x⁴ + 25

2) f

(

x

)

^{= 2}

x⁵ − 5 f'

(

x

)

^{= − 2 ⋅ 5x}

4

(

x⁵ − 5

)

²

= − 10x⁴ x¹⁰ − 10x⁵ + 25

3) f

(

x

)

^{= 5}

4x³ + 4 f'

(

x

)

^{= − 5 ⋅ 12x}

2

(

⁴x³ + 4

)

²

= − 15x² 4x⁶ + 8x³ + 4

4) y =

4x³ − 3x²

4x⁵ − 4 dy

dx =

(

⁴x⁵ − 4

)(

¹²x² − 6x

)

⁻

(

⁴x³ − 3x²

)

^{⋅ 20}x⁴

(

⁴x⁵ − 4

)

²

=

−8x⁷ + 9x⁶ − 12x² + 6x 4x¹⁰ − 8x⁵ + 4

5) y =

3x⁴ + 2

3x³ − 2 dy

dx =

(

³x³ − 2

)

^{⋅ 12}x³ −

(

³x⁴ + 2

)

^{⋅ 9}x²

(

³x³ − 2

)

²

=

9x⁶ − 24x³ − 18x² 9x⁶ − 12x³ + 4

6) y =

4x⁵ + 2x²

3x⁴ + 5 dy

dx =

(

³x⁴ + 5

)(

²⁰x⁴ + 4x

)

⁻

(

⁴x⁵ + 2x²

)

^{⋅ 12}x³

(

³x⁴ + 5

)

²

=

12x⁸ − 12x⁵ + 100x⁴ + 20x 9x⁸ + 30x⁴ + 25

7) y =

4x⁵ + x² + 4 5x² − 2 dy

dx =

(

⁵x² − 2

)(

²⁰x⁴ + 2x

)

⁻

(

⁴x⁵ + x² + 4

)

^{⋅ 10}x

(

⁵x² − 2

)

²

=

60x⁶ − 40x⁴ − 44x 25x⁴ − 20x² + 4

8) y =

3x⁴ + 5x³ − 5 2x⁴ − 4 dy

dx =

(

²x⁴ − 4

)(

¹²x³ + 15x²

)

⁻

(

³x⁴ + 5x³ − 5

)

^{⋅ 8}x³

(

²x⁴ − 4

)

²

=

−5x⁶ − 4x³ − 30x² 2x⁸ − 8x⁴ + 8

-1-

©w M2K0G1G3Y PKKuntcat BS4oMfwt2wqa6rDe1 FLALGCh.1 u 0A6l3ly BrtikgvhitHsn yrdeEs6e2rkvbedds.K Z 3MHa2ddem JwsiGtNhv VIcnVfjisnwiKtveH eCDaWlWcruelauNsw.F Worksheet by Kuta Software LLC

9) y =

x³ − x² − 3

x⁵ + 3 dy

dx =

(

x⁵ + 3

)(

³x² − 2x

)

⁻

(

x³ − x² − 3

)

^{⋅ 5}x⁴

(

x⁵ + 3

)

²

=

−2x⁷ + 3x⁶ + 15x⁴ + 9x² − 6x x¹⁰ + 6x⁵ + 9

10) y =

x⁴ + 6 3 − 4

x⁻⁴

dy dx

=

(

^{3 − 4}

x⁻⁴

)

^{⋅ 4}x³ −

(

x⁴ + 6

)

^{⋅ 16}

x⁻⁵

(

^{3 − 4}

x⁻⁴

)

²

=

12x¹¹ − 32x⁷ − 96x³ 9x⁸ − 24x⁴ + 16

11) y =

4x⁴ − 4x² + 5

2

x

5

3 + 3

dy dx

=

(

2

x

5

3 + 3

) (

¹⁶x³ − 8x

)

⁻

(

⁴x⁴ − 4x² + 5

)

^⋅

10 3

x

2 3

(

2

x

5 3 + 3

)

²

=

56

x

14

3 + 144x³ − 8

x

8

3 − 72x − 50

x

2 3

12

x

10 3 + 36

x

5 3 + 27

Critical thinking question:

12) A classmate claims that

(

g^f

)

^{' =} g^f'^' for any functions f and g. Show an example that proves your classmate wrong.

Many answers. Ex: f = 4, g = 2x, − 2 x²

≠ 0

-2-

Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com