As in single variable calculus, there is a multivariable chain rule. The version with several variables is more complicated and we will use the tangent approximation and total differentials to help understand and organize it.
Also related to the tangent approximation formula is the gradient of a function. The gradient is one of the key concepts in multivariable calculus. It is a vector field, so it allows us to use vector techniques to study functions of several variables. Geometrically, it is perpendicular to the level curves or surfaces and represents the direction of most rapid change of the function. Analytically, it holds all the rate information for the function and can be used to compute the rate of change in any direction.