- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Functions of two variables and partial derivatives practice question 6
Let . Find at .
With held fixed, is a quotient of two functions of . Which rule applies, and in what order does its subtraction go?
Hold fixed and use the quotient rule in . The top and the bottom each have -derivative 1.
The numerator simplifies: .
A fraction, or a decimal to four places.
Theory UBC Math 100 §15.1Key results
Worked solution
Strategy
With y held constant, f is a quotient of two functions of x. Apply the quotient rule, simplify the numerator, then substitute.
Working
Bottom times the derivative of the top, minus top times the derivative of the bottom, over the bottom squared.
The x terms cancel: x + 2y − x + y = 3y.
Substitute x = 2 and y = 1.
Answer
The answer is three sixteenths, 0.1875.
Check
With y = 1, f = (x − 1)/(x + 2) = 1 − 3/(x + 2), whose slope 3/(x + 2)² is 3/16 at x = 2, positive as f climbs toward 1.
Where this goes wrong
Reversing the subtraction in the quotient rule and getting −3/16. The order matters: the bottom times the derivative of the top comes first.
Leans on
The quotient rule, from Differentiation rules
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