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 …