# SOLUTION: Calculus Algebra Worksheet – Studypool

SOLUTION: Calculus Algebra Worksheet – Studypool.

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Derivatives

1. Find derivatives:

a. y = ( x + 7 ) ( x − 9 )

3

4

Differentiating both sided with respect to x and using product rule on RHS

dy d

3

4

=

x + 7) ( x − 9)

(

dx dx

3 d

4

4 d

3

= ( x + 7)

( x − 9) + ( x − 9) ( x + 7)

dx

dx

= 4 ( x + 7 ) ( x − 9) + 3( x − 9) ( x + 7 )

3

3

4

2

= ( x + 7 ) ( x − 9 ) 4 ( x + 7 ) + 3 ( x − 9 )

2

3

= ( x + 7 ) ( x − 9 ) ( 7 x + 1)

2

b. y =

3

x2 − 5

x −5

Differentiating both sided with respect to x and then using quotient rule on RHS

dy d x 2 − 5

=

dx dx x − 5

=

=

=

( x − 5) ( 2 x ) − ( x 2 − 5 ) (1)

( x − 5)

2

(2 x 2 − 10 x) − ( x 2 − 5 )

( x − 5)

x 2 − 10 x + 5

( x − 5)

2

2

c.

y = ( x3 + 4 ) 4 x 2 + 2 x − 5

Differentiating using product rule

dy d 3

=

x + 4) 4×2 + 2x − 5

(

dx dx

= 3x 2 4 x 2 + 2 x − 5 + ( x3 + 4 )

=

=

=

d. y =

3

8x + 2

2 4 x2 + 2 x − 5

6 x 2 ( 4 x 2 + 2 x − 5 ) + 8 x 4 + 2 x 3 + 32 x + 8

2 4 x2 + 2 x − 5

32 x 4 + 14 x3 − 30 x 2 + 32 x + 8

2 4 x2 + 2 x − 5

16 x 4 + 7 x3 − 15 x 2 + 16 x + 4

4 x2 + 2 x − 5

2 x2 + 5

( x + 3)

4

Differentiating both sides with respect to x

1

2

3

2

x

+

5

)

dy d (

=

4

dx dx ( x + 3)

−2

1

4 1

3

2

3 4x − 2×2 + 5 3 4 x + 3

x

+

3

2

x

+

5

(

) (

( ) (

(

)

)

)

3

=

8

( x + 3)

4 x ( x + 3)

=

3( 2 x 2 + 5)

4

23

− 4 ( x + 3) ( 2 x 2 + 5 )

3

( x + 3)

8

4 x ( x + 3) − 12 ( 2 x 2 + 5 )

( x + 3)

23

2

3

2

x

+

5

(

)

3

=

=

=

13

( x + 3)

8

4 ( −5 x 2 + 3 x − 15 )

3 ( x + 3) ( 2 x 2 + 5 ) 2 3

5

−4 ( 5 x 2 − 3 x + 15 )

3 ( x + 3) ( 2 x 2 + 5 ) 2 3

5

2. 5 x3 y 4 +

3y2

= 5 xy

4 x3

d 3 4 3y2

d

5 x y + 3 = 5 ( xy …

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