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MATH 171 - Derivative Worksheet
Differentiate these for fun, or practice, whichever you need. The given answers are not simplified.
- f (x) = 4x
5 − 5 x
4
- f (x) = e
x sin x 3. f (x) = (x
4
− 1
- f (x) = 3x
2 (x
3
7
- f (x) = cos
4 x − 2 x
2
- f (x) =
x
1 + x^2
- f (x) =
x 2 − 1
x
- f (x) = (3x
2 )(x
1 (^2) ) 9. f (x) = ln(xe^7 x)
- f (x) =
2 x
4
2 − 1
x 2
- f (x) = (x
3 )
5
2 − x 12. f (x) = 2x −
x
- f (x) =
4(3x − 1) 2
x 2
- f (x) =
x^2 + 8 15. f (x) =
x √ 1 − (ln x) 2
- f (x) =
(3x 2 − π) 4
- f (x) =
(3x
2 − πx)
4
- f (x) =
x
(x 2
3 x) 5
- f (x) = (xe
x )
π
- f (x) =
[ arctan(2x)
] 10
- f (x) = (e
2 x
1 2
- f (x) = (x
6
5 (4x + 7)
3
- f (x) = (7x +
x^2 + 3)
6
- f (x) =
1 x
1 x^2
x − 1
- f (x) =
3
x^2 −
x 3
- f (x) =
√ 2 x + 5
7 x − 9
- f (x) =
sin x
cos x
- f (x) = e x (x 2 + 3)(x 3 + 4) 29. f (x) =
5 x 2 − 7 x
x 2
- f (x) =
[ ln(5x 2
3
- f (x) = ln(5x
2
3
- f (x) = cot(6x) 33. f (x) = sec
2 x · tan x
- f (x) = arcsin(
x ) 35. f (x) = tan(cos x) 36. f (x) = [(x
2 − 1)
5 − x]
3
- f (x) = sec x · sin(3x) 38. f (x) =
(x − 1) 3
x(x + 3)^4
- f (x) = log 5 (3x
2
In problems 40 – 42, find
dy
dx
. Assume y is a differentiable function of x.
- 3y = xe
5 y
- xy + y 2
sin y
y^2 + 1
= 3x
If f and g are differentiable functions such that f (2) = 3 , f
′ (2) = −1 , f
′ (3) = 7 , g(2) = − 5
and g ′ (2) = 2 , find the numbers indicated in problems 43 – 48.
- (g − f )
′ (2) 44. (f g)
′ (2) 45.
( f
g
)′
- (5f + 3g)
′ (2) 47. (f ◦ f )
′ (2) 48.
( f
f + g
)′
Answers: Absolutely not simplified ... you should simplify more.
- f
′ (x) = 20x
4 − 20 x
3
- f
′ (x) = e
x cos x + (sin x)e
x
- f
′ (x) = −1(x
4
− 2 (4x
3
′ (x) = 3x
2 · 7(x
3
6 (3x
2 ) + (x
3
7 · 6 x
- f ′ (x) = 4(cos x)
3 (− sin x) − 4 x 6. f ′ (x) =
(1 + x 2 )(1) − x(2x)
(1 + x^2 )^2
- f
′ (x) = 1 + x
− 2 (Simplify f first.) 8. f
′ (x) = 3 ·
x
3 (^2) (Simplify f first.)
- f ′ (x) =
x
- 7 (Simplify f first.) 10. f ′ (x) = 4x + 0 + 2x
− 3 (Simplify f first.)
- f
′ (x) = x
3 ·
(2 − x)
− 4 (^5) (−1) + (2 − x)
1 (^5) (3x^2 ) 12. f ′(x) = 2 + 2x
− 3 2
- f ′ (x) =
(x
2
x )
[ 4 · 2(3x − 1)(3)
] − 4(3x − 1)
2 (2x + 7
x ln 7)
(x^2 + 7x)^2
- f ′ (x) =
(x
2
− 1 (^2) (2x)
- f ′ (x) =
( 1 − (ln x)
2
) 1 2 (1) − x ·
1 2
( 1 − (ln x)
2
) − 1 2
( − 2(ln x) ·
1 x
)
1 − (ln x)^2
- f ′ (x) = −24(3x
2 − π)
− 5 (6x)
- f ′ (x) =
[ 4(3x
2 − πx)
3 (6x − π)
]
- f ′ (x) =
(x 2
3 x) 5 (1) − x
[ 5(x 2
3 x) 4
( 2 x +
1 2 (3x)
− 1 (^2) · 3
)]
(x^2 +
3 x)^10
- f ′ (x) = π(xe
x )
(π−1)
[ xe
x
x
]
- f ′ (x) = 10
[ arctan(2x)
] 9 ·
1 + (2x)^2
- f ′ (x) =
(e 2 x
− 1 (^2) (e^2 x^ · 2 + 0) 22. f ′(x) = (x^6 + 1)^5
[ 3(4x + 7) 2 (4)
]
[ 5(x 6
4 (6x 5 )
]
- f ′ (x) = 6(7x +
x 2
5
( 7 +
(x
2
− 1 (^2) · 2 x
)
- f ′ (x) =
(x − 1)(−x
− 2 − 2 x
− 3 ) − (x
− 1
− 2 )(1)
(x − 1)^2
- f
′ (x) =
x
− 1 (^3) +
x
− 5 (^2) 26. f ′(x) =
( 2 x + 5
7 x − 9
) −^1 2
[ (7x − 9)(2) − (2x + 5)(7)
(7x − 9) 2
]
- f ′ (x) = sec
2 x 28. f ′ (x) =
[ e
x (x
2
] (3x
2 ) + (x
3
[ e
x (2x) + (x
2
x
]
- f
′ (x) =
(x
2
2 − 7 x)(2x)
(x 2
2
- f
′ (x) = 3
[ ln(5x
2
] 2 ·
5 x 2
(10x + 0)
- f ′ (x) =
(5x 2
3
[ 3(5x
2
2 (10x + 0)
]
- f ′ (x) = − csc
2 (6x) · 6
- f ′ (x) = sec
2 x(sec
2 x) + tan x
[ 2 · sec x(sec x tan x)
]
- f ′ (x) =
√ 1 − (2x)^2
x ln 2
- f ′ (x) =
( sec
2 (cos x)
) (− sin x) 36. f ′ (x) = 3
[ (x
2 − 1)
5 − x
] 2 ( 5(x
2 − 1)
4 · 2 x − 1
)
- f
′ (x) = sec x
( cos(3x) · 3
)
( sec x tan x
)
- f ′ (x) =
x(x + 3) 4
[ 3(x − 1) 2 (1)
] − (x − 1) 3
[ x · 4(x + 3) 3 (1) + (x + 3) 4 (1)
]
x^2 (x + 3)^8
- f ′ (x) =
(3x^2 + 4x) · ln 5
· (6x + 4) 40.
dy
dx
e 5 y
3 − 5 xe^5 y
dy
dx
− 3 x
2 − y
x + 2y
dy
dx
3(y
2
2
(y 2
Chain Rule Worksheet
Find the derivative of each function.
2 3 f ( ) x = (2 x − 5 ) x 2.
3 f ( ) x = 5 x − 2 x
- y = 3sin( x − 3) 4.
2 y = −2 cos( x +2)
2 2 g x ( ) = sin (3 x ) 6.
3 2 h x ( ) = sec ( x −5)
3 2 5 ( ) 3
x f x x e
− = 8.
2 23 ( ) 5
x x g x x e
= −
2 2 y = 3 x 4 x − 5 x + 1 10.
h t = t t − t
3 3 2
y x x
4 3
g t t t
g m ( ) = sin(cos( m )) 14. f ( ) x =cos(tan x )
3 2 4 h x ( ) = x + 2( x − 1) 16.
2 2 3 h m ( ) = m + 1( m +1)
2 3 2
t f t t
3 4 3
t f t t
7 4 5 h x ( ) = (2 x + 5) (3 x − 8) 20.
2 3 g n ( ) = (3 x − 2)(4 x +1)
2 3 f t ( ) = csc ( t ) 22.
4 2 f t ( ) =cot (2 t )
23 2 ( )
x x h x e
− = 24.
4 23 ( )
x x f x e
−
3 2
x h x x
3
4 2
s f s s s
sin^3 ( ) 5
x f x = 28.
4 ( ) 2
ex f x =
