Questions: Interpreting Partial Derivatives as Rates of Change

Question 1

If f(x, y) = x² + xy and ∂f/∂x = 2 at the point (1, 0), what does this number mean geometrically?

• A. The surface z = f(x, y) has a slope of 2 in every direction from (1, 0, 1)
• B. The tangent plane at (1, 0, 1) rises 2 units in both the x and y directions
• C. The cross-sectional curve formed by fixing y = 0 and varying x has slope 2 at x = 1
• D. f increases by exactly 2 regardless of which direction you step from (1, 0)

Answer: (C)

∂f/∂x at a point measures the slope of the curve formed by slicing the surface with the plane y = 0 (holding y fixed at its current value). That cross-section gives z = f(x, 0) = x², a function of x alone, whose ordinary derivative at x = 1 is 2. Options A and D are wrong because the partial derivative gives the rate only in the x-direction, not all directions.

Question 2

The function f(x, y) = x³ + y² has ∂f/∂x = 3x² and ∂f/∂y = 2y. At the point (0, 1), ∂f/∂x = 0. A student concludes that f is not changing at this point. What is wrong?

• A. ∂f/∂x = 0 signals a critical point, meaning the function must be changing more rapidly near (0, 1)
• B. ∂f/∂x = 0 means f is not changing in the x-direction only; ∂f/∂y = 2(1) = 2 ≠ 0, so f is still increasing in the y-direction
• C. The student should have computed the total derivative, not just one partial derivative
• D. ∂f/∂x = 0 cannot be correct because f is a cubic function

Answer: (B)

∂f/∂x = 0 only says there is no rate of change in the x-direction — the cross-sectional slice y = 1 has a flat tangent at x = 0. But ∂f/∂y = 2y = 2 at (0, 1), meaning f is increasing at rate 2 in the y-direction. Each partial derivative is a rate in one coordinate direction; zero in one direction does not mean zero everywhere.

Question 3

The partial derivative ∂f/∂x at a point gives the rate of change of f in nearly every direction from that point, not just along the x-axis.

• T. True
• F. False

Answer: False

∂f/∂x measures only the rate of change as you move parallel to the x-axis, with y held fixed. The rate of change in an arbitrary direction requires the directional derivative, which is ∇f · u where u is the unit direction vector. Partial derivatives are the building blocks — the coordinate-axis rates — not the complete picture of directional change.

Question 4

∂f/∂x at (a, b) equals the slope of the tangent line to the curve z = f(x, b) — the cross-section of the surface obtained by fixing y = b — evaluated at x = a.

• T. True
• F. False

Answer: True

Holding y = b turns f(x, y) into a single-variable function f(x, b). The partial derivative ∂f/∂x is exactly the ordinary derivative of this function at x = a, which is the slope of its tangent line. This is the precise geometric meaning: a partial derivative is a slope along a coordinate cross-section of the surface.

Question 5

What does it mean geometrically to 'hold y fixed' when computing ∂f/∂x, and why does this reduce the problem to a single-variable calculation?

Model answer: Holding y = b fixed means restricting attention to the vertical plane y = b, which slices the surface z = f(x, y) into a curve z = f(x, b). Along this curve, y is a constant, so f depends only on x. The partial derivative ∂f/∂x is then the ordinary derivative of this single-variable function — the slope of the tangent to the cross-sectional curve. The multivariable surface is temporarily reduced to a 2D curve by this slicing operation, making familiar single-variable calculus directly applicable.

This is the key interpretive link: partial differentiation is not a new operation but ordinary differentiation applied along a carefully chosen slice of a higher-dimensional surface. The 'holding fixed' operation is what selects which slice to use.