Partial Derivatives: Step-by-Step Guide
A partial derivative measures how a multivariable function changes as one variable varies while all others are held constant. The partial derivative of f(x, y) with respect to x is denoted f_x or df/dx, and is computed by treating y as a constant and differentiating with respect to x using the usual single-variable rules.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Identify the function and the variable of differentiation. Decide which variable you are differentiating with respect to. All other variables are treated as constants.\[\frac{\partial f}{\partial x} \quad \text{or} \quad f_x(x,y)\]
- Treat the other variables as constants. Mentally replace the held-constant variables with concrete numbers (like a or k) so the function reduces to a single-variable problem.\[f(x,y) \to g(x) \text{ with } y \text{ fixed}\]
- Differentiate using single-variable rules. Apply power, product, quotient, chain, and trig rules as you would for a single-variable function.\[f_x = \frac{d}{dx}\bigl[g(x)\bigr]\]
- Repeat for each variable as needed. Compute f_y by holding x constant. Higher-order partials follow by differentiating again (e.g., f_xy = (f_x)_y).\[f_y,\; f_{xx},\; f_{xy},\; f_{yy}\]
- Evaluate at a point if requested. Plug in specific values of x and y into the partial derivative.\[f_x(a,b) = \left.\frac{\partial f}{\partial x}\right|_{(a,b)}\]
Worked example
Find f_x and f_y for f(x, y) = x^3 y^2 + sin(xy), and evaluate f_x at (1, 2).
Step 1: To find f_x, treat y as a constant.
Step 2: Differentiate term by term. d/dx[x^3 y^2] = 3x^2 y^2 (since y^2 is constant). d/dx[sin(xy)] = cos(xy) * y by chain rule.
Step 3: f_x = 3x^2 y^2 + y cos(xy).
Step 4: For f_y, treat x as constant. d/dy[x^3 y^2] = 2 x^3 y. d/dy[sin(xy)] = cos(xy) * x. So f_y = 2 x^3 y + x cos(xy).
Step 5: Evaluate f_x at (1, 2): f_x(1, 2) = 3(1)^2(2)^2 + 2 cos(2) = 12 + 2 cos(2).
Verification: Numerically, 2 cos(2) is approximately -0.832, so f_x(1, 2) is approximately 11.17. The dominant 3x^2 y^2 term contributes 12, consistent with our answer.
Tips and common mistakes
- When taking a partial with respect to x, mentally pretend y is a constant like 5; this simplifies the rules.
- All single-variable derivative rules (power, product, quotient, chain) apply unchanged to partials.
- The chain rule is essential: d/dx[f(xy)] = f'(xy) * y, not just f'(xy).
- Mixed partials f_xy and f_yx are equal whenever both are continuous (Clairaut's theorem).
- Use subscript notation f_x for compactness in intermediate steps.
- Be careful with terms that are constant in the variable you're differentiating with respect to - their partial is 0.
- When in doubt, write out the function with the held variable replaced by a letter constant, then differentiate.
- Always double-check by computing both f_x and f_y; they should be different (unless the function is symmetric).