Partial Derivatives 101: A Beginner's Guide

If you've learned single‑variable calculus, partial derivatives extend the same ideas to functions with multiple variables. We'll keep it approachable and practical: what they mean, how to compute them, and how they connect to gradients and tangent planes. Try anything here inside MathAI GPT to get instant step‑by‑steps.

1) What is a Partial Derivative?

Suppose f(x, y) depends on two variables. The partial derivative ∂f/∂xmeasures the rate of change of f with respect to x while keeping y fixed (treat y like a constant). Similarly for ∂f/∂y.

• Read ∂ as “partial.”
• Notation options: f_x, ∂f/∂x, or D_x f.

2) Fast Rules You Already Know (Treat Others as Constants)

• Power rule: ∂/∂x (x^n) = n x^{n-1}; constants multiply through.
• Sum / difference: differentiate term‑by‑term.
• Product rule (two terms in x): (uv)_x = u_x v + u v_x.
• Chain rule: if u = u(x, y), then f_x = f_u u_x + f_v v_x for appropriate intermediate variables.
• Exponential / trig: as usual, but only the variable you're differentiating is “active.”

3) Worked Examples

4) Mixed Partials and Clairaut's Theorem (Idea Only)

Mixed partials differentiate in two orders, e.g., f_{xy} = ∂/∂y (f_x) andf_{yx} = ∂/∂x (f_y). If f is “nice enough” (continuous partials nearby), these are equal: f_{xy} = f_{yx}.

5) Directional Derivatives and the Gradient

For a unit direction vector \u005E{u} (pronounced “u‑hat”), the directional derivative isD_u f = ∇f · \u005E{u}. The gradient ∇f = ⟨f_x, f_y⟩ points toward the steepest increase; its magnitude |∇f| is the steepest rate.

6) Gradient and Tangent Plane (Why Partials Matter)

The gradient collects first partials: ∇ f = ⟨ f_x, f_y ⟩. At a point (a, b), the linear approximation (tangent plane to the surface z = f(x, y)) is

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

This is useful for quick estimates and optimization methods.

7) Critical Points and the Second Derivative Test

Critical points satisfy f_x = 0 and f_y = 0. Let the Hessian beH = [[f_{xx}, f_{xy}], [f_{yx}, f_{yy}]] and defineD = f_{xx} f_{yy} - (f_{xy})^2.

• D > 0, f_{xx} > 0 → local minimum
• D > 0, f_{xx} < 0 → local maximum
• D < 0 → saddle point
• D = 0 → test inconclusive

Example: f(x,y)=x^2+y^2. The only critical point is (0,0). Here f_{xx}=2, f_{yy}=2, f_{xy}=0 → D=4>0 and f_{xx}>0, so it is a local (and global) minimum.

8) Multivariable Chain Rule (Quick Pattern)

Total differential: df = f_x dx + f_y dy. If x=x(t), y=y(t), thendf/dt = f_x x'(t) + f_y y'(t).

9) Tips and Common Pitfalls

• When taking ∂/∂x, treat y as a constant everywhere (and vice‑versa).
• Write one line per step for chain/product rules to avoid dropped factors.
• Check special points (zeros in factors, undefined spots) before simplifying too aggressively.
• When rules mix: do chain rule for inner functions first, then apply product rule around them.
• Mind domains: logs need positive inputs; radicals need non‑negatives (unless complex analysis is intended).

10) Practice (Answers Hidden)

1. f(x, y) = x^3 + x y^2 - 4y: find f_x, f_y.
2. f(x, y) = e^{x^2 y}: find f_x, f_y.
3. f(x, y) = (x^2 + y)\,ln(x): find f_x (product + chain).
4. Critical point classification: f(x,y)=x^2- y^2. Find critical points and use D = f_{xx} f_{yy} - (f_{xy})^2 to classify.

Next Step

Open the partial derivative calculator and paste a function you're studying (e.g., f(x,y) = x e^{xy} + y^3). Ask for: the first partials, the gradient at a point you choose, and the tangent plane at that point.

Bonus: Ask for a quick visual sketch description of level curves near your point.