Partial Derivatives and Gradients

GRE Math Subject · Axiom Academy

LESSON Partial Derivatives and Gradients Multivariable rates of change, directional derivatives, the Jacobian, the Hessian, and Lagrange multipliers Definition: For f(x, y), the partial derivative with respect to x is: Treat all other variables as constants and differentiate with respect to the target variable using single-variable rules. Example: Let f(x, y) = 3x² + 2xy + y³. Geometric interpretation: The partial derivative gives the slope of f in the x-direction while y is held fixed -- it is the slope of the trace curve obtained by slicing the surface z = f(x, y) with a plane y = constant. Higher-Order Partials and Clairaut's Theorem Clairaut's Theorem: If the mixed second partial derivatives are continuous on a region, then: This is frequently tested on the GRE as a quick shortcut -- you only need to compute one mixed partial. Definition: For f : R² → R: Key properties of the gradient: points in the direction of steepest increase of f. gives the maximum rate of increase . is perpendicular to level curves (2D) or level surfaces (3D) of f. If at a point, the function has no preferred direction of increase there (candidate for extremum). Example: f(x, y) = x² + y² At (1, 1): , pointing radially outward from the origin. The level curves are circles x² + y² = c, and the gradient is perpendicular to them. Definition: The rate of change of f at point P in the direction of a unit vector u is:

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