Partial Derivatives

Partial derivatives extend differentiation to functions of multiple variables by measuring the rate of change with respect to one variable while holding others constant. This topic covers basic and higher-order partial derivatives, the multivariable chain rule, gradient vectors, and optimization with critical points and the second derivative test.

Partial derivatives power machine learning (gradient descent), describe temperature distributions in physics (heat equation), and enable sensitivity analysis in economics and engineering.

Practice Tips

• 1When computing a partial derivative with respect to x, treat every other variable as a constant, just like single-variable differentiation.
• 2For the gradient, compute all first partial derivatives and assemble them into a vector; this vector always points in the direction of steepest increase.
• 3In multivariable optimization, use the second partial derivative test (D = f_xx * f_yy - f_xy^2) to classify critical points as maxima, minima, or saddle points.