- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Functions of two variables and partial derivatives practice question 3
Let . Find .
In , which letter is held fixed? Go through the four terms and ask which of them change when only changes.
Treat as a constant and differentiate in : becomes , and a term with no in it, such as , differentiates to 0.
A whole number.
Theory UBC Math 100 §15.1Key results
Worked solution
Strategy
A partial derivative in x is an ordinary derivative in x with y frozen at a constant. Differentiate term by term, then substitute (1, 2).
Working
With y constant, y² and 3y are coefficients, and 2y³ is a constant term.
Substitute x = 1 and y = 2.
Answer
The answer is 10.
Check
Freezing y = 2 first gives f(x, 2) = 4x³ − 2x + 16, whose ordinary derivative at x = 1 is 12 − 2 = 10.
Where this goes wrong
Treating y as if it changes with x: differentiating 2y³ to 6y², or using the product rule on x³y². In f_x the variable y is a fixed number, so terms in y alone vanish and powers of y are just coefficients.
Leans on
The power rule, from Differentiation rules
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