Introduction to Calculus
Higher Order Derivatives
• The derivative [pic] is called the first derivative of y with respect to x.
• The first derivative may itself be a differentiable function of x. If so, its derivative,
[pic] is called the second derivative of y with respect to x.
• The names continue, but the multiple prime notation begins to lose its usefulness after three primes. Therefore the notation becomes [pic] which implies the nth derivative of y with respect to x.
For #1-2: Find each derivative of the functions below.
1. [pic] [pic]
[pic] [pic]
2. [pic] [pic] [pic]
[pic] [pic]
In general, what is the relationship between the degree of a polynomial function and the degree of its successive derivatives?
For #3-7: Find the first four derivatives of each function below.
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic]
Did the relationship you found for polynomial degrees and the number of non-zero derivatives hold for other types of functions?
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
8. If [pic], then [pic]. _____
9. If [pic] then [pic]. _____
10. If [pic] are zero and [pic], then [pic]. _____
11. If [pic] is an nth-degree polynomial, then [pic]. _____
12. The second derivative represents the rate of change of the first derivative. _____
13. If the velocity of an object is constant, then its acceleration is zero. _____
14. Suppose that [pic]and [pic], [pic], [pic], [pic], [pic], and [pic]. Find [pic].
15. A table of values for f, g, [pic], and [pic] is given.
|x |[pic] |[pic] |[pic] |[pic] |
|1 |3 |2 |4 |6 |
|2 |1 |8 |5 |7 |
|3 |7 |2 |7 |9 |
a) If [pic], find [pic].
b) If [pic], find [pic].
c) If [pic], find [pic].
d) If [pic], find [pic].
More practice from the other worksheet:
16. [pic] 17. [pic]
18. [pic] 19. [pic]
20. [pic] 21. [pic]
22. [pic]
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