- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Functions of two variables and partial derivatives practice question 4
Let . Find .
With held fixed, look at each term in turn: which factors are just numbers, and which part actually changes with ?
Treat as a constant. Then is a constant coefficient on , is a constant coefficient on , and is a constant term.
A formula in x and y. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §15.1Key results
Worked solution
Strategy
A partial derivative in x is an ordinary derivative in x with y frozen. Differentiate term by term, treating every factor that contains only y as a number.
Working
With y fixed, e2y is a constant multiple of x.
Power rule on x², with −3y as the coefficient; a term in y alone is a constant.
Add the three pieces.
Answer
The answer is f sub x equals e to the 2y, minus 6xy.
Check
Along y = 1, f = e²x − 3x² + 1, whose slope e² − 6x is exactly what e2y − 6xy gives at y = 1.
Where this goes wrong
Differentiating y⁴ to 4y³. In f_x the variable y is a fixed number, so a term in y alone has x-derivative 0, and a factor in y alone, such as e2y, is just a coefficient.
Leans on
The power rule, from Differentiation rules
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