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# The Product and Quotient Rules Practice Questions

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Updated on Oct 1, 2011

To review these concepts, go to The Product and Quotient Rules Study Guide.

## The Product and Quotient Rules Practice Questions

Differentiate the following.

1. f(x) = x2cos(x)
2. y = 8t3et
3. y = sin(x)cos(x)
4. g(x) = 3x2ln(x) – 5x4 + 10
5. h(u) = ueueu
6. k(x) = sin(x) + x4x2sin(x)
7. y = 8ln(x)sin(x) + cos(x)
8. h(t) = tsin(t) – tcos(t)
9. y = 5x3xln(x)
10. f(x) = sin2(x) = sin(x) · sin(x)
11. y = xexsin(x)
12. g(x) = 3x4ln(x)cos(x)
13. What is the slope of the tangent line to f(x) = x2ex + x + 2 at (0,2)?
14. Find the equation of the tangent line to y = xsin(x) at x = π.

Differentiate the following.

1.
2. Find the second derivative of
3. What is the slope of the tangent line to

Differentiate the following.

1. y = csc(x)
2. y = cot(x)
3. f(x) = xtan(x)

### Solutions

1. f'(x) = 2xcos(x) – sin(x) · x2
2. = 24t2et + et · 8t3 = 8t2et(3 + t)
3. = cos2(x) – sin2(x)
4. g'(x) = 6xln(x) + (3x2) – 20x3 = 6xln(x) + 3x – 20x3
5. h'(u) = 1 · eu + eu · ueu = ueu
6. 6. k'(x) = cos(x) + 4x3 – (2xsin(x) + cos(x)x2) = cos(x) + 4x3 – 2xsin(x) – x2cos(x)
7. = + 8ln(x)cos(x) – sin(x)
8. h'(t) = (sin(t) + cos(tt) – (cos(t) – sin(t) · t) = sin(t) + tcos(t) – cos(t) + tsin(t)
9. = 15x2 – (1 · ln(x) + · x) = 15x2 – ln(x) – 1
10. f'(x) = cos(x)sin(x) + cos(x)sin(x) = 2cos(x)sin(x)
11. = exsin(x) + (exsin(x) + cos(x)ex · x
12. g'(x) = 12x3ln(x)cos(x) + ((x) – sin(x)ln(x)) · 3x4 =12x3ln(x)cos(x) + 3x3cos(x) – 3x4sin(x)ln(x)
13. f'(0) = 2(0)e0 + e(0)2 + 1 = 1) so the slope is 1.
14. y = – π (x – π) = –πx + π2
15. h'(x) =
16. =
17. f'(x) =
18. = =
19. f'(x) = =
20. g'(t) =
21. =
22. g'(u) =
23. =
24. h'(t) = =
25. = =
26. f'(x) =
27. =
28. The slope is f'(5) = .
29. (csc(x)) = () = = –csc(x)cot(x)
30. (cot(x)) = ()() = = –csc2(x)
31. f '(x) = tan(x) + x · sec2(x)
32. g'(x) =

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