Gradient: Step-by-Step Guide
The gradient of a scalar function f is the vector of its partial derivatives. It points in the direction of greatest increase of f at each point, and its magnitude equals the maximum rate of change. It is also normal to the level curves (or surfaces) of f.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Identify the variables. Determine whether f is a function of two, three, or more variables. The gradient has one component per variable.\[f(x,y),\; f(x,y,z),\; f(x_1,\dots,x_n)\]
- Compute each partial derivative. For each variable, hold the others constant and differentiate.\[f_x,\; f_y,\; f_z,\;\dots\]
- Assemble the gradient vector. Write the partials as the components of a vector.\[\nabla f = \langle f_x,\, f_y,\, f_z \rangle\]
- Evaluate at a point if asked. Substitute the coordinates of the point into each component.\[\nabla f(P) = \langle f_x(P),\, f_y(P),\, f_z(P) \rangle\]
- Interpret geometrically. The gradient points in the direction of steepest ascent and is perpendicular to the level set through P.\[\nabla f(P) \perp \{f = f(P)\}\]
Worked example
Find the gradient of f(x, y) = x^2 + 3xy at the point (1, 2).
Step 1: f is a function of two variables, so the gradient has two components.
Step 2: Compute partials. f_x = 2x + 3y. f_y = 3x.
Step 3: Write the gradient: grad f = (2x + 3y, 3x).
Step 4: Evaluate at (1, 2): grad f(1, 2) = (2(1) + 3(2), 3(1)) = (8, 3).
Step 5: The gradient is (8, 3), pointing in the direction of steepest ascent at (1, 2). The maximum rate of increase is sqrt(64 + 9) = sqrt(73) ~ 8.54.
Verification: Both components are positive, consistent with f increasing in both x and y near (1, 2).
Tips and common mistakes
- The gradient is a vector; the partials are scalars.
- Number of components equals number of variables.
- grad f points in the direction of fastest increase; -grad f points in the direction of fastest decrease.
- |grad f| is the maximum rate of change in any direction.
- grad f is perpendicular to the level curve f = c at every point.
- Use the gradient to set up the equation of the tangent plane to a level surface: grad F dot (r - r0) = 0.
- If grad f(P) = 0, then P is a critical point - candidate for max, min, or saddle.
- For composite functions, use the multivariable chain rule rather than expanding.