Questions: Partial Derivatives: Definition and Computation
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
Given f(x, y) = x²y + 3y², what is ∂f/∂x?
A2xy + 6y
B2xy
Cx² + 6y
D2x + 3y²
When computing ∂f/∂x, y is treated as a constant. Differentiating x²y with respect to x gives 2xy (y acts as a constant coefficient). The term 3y² has no x in it, so its partial derivative with respect to x is 0. Result: 2xy.
Question 2 True / False
The partial derivative ∂f/∂x of f(x, y) = x² + y² gives the same result as the total derivative df/dx when y is a function of x.
TTrue
FFalse
Answer: False
When y depends on x, the total derivative df/dx includes an additional term: df/dx = 2x + 2y·(dy/dx) by the chain rule. The partial derivative ∂f/∂x = 2x treats y as constant regardless of whether y actually depends on x. They are equal only when dy/dx = 0.
Question 3 Short Answer
Given f(x, y, z) = x²yz + sin(y), what is ∂f/∂y?
Think about your answer, then reveal below.
Model answer: x²z + cos(y)
Hold x and z constant. Differentiating x²yz with respect to y gives x²z (since x² and z are constants). Differentiating sin(y) with respect to y gives cos(y). Result: x²z + cos(y).