Partial Derivatives, the Chain Rule and Directional Derivatives: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 4, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
The derivative of a multivariable function with respect to one variable while all other variables are held fixed, measuring the rate of change along a coordinate direction.
Notation f_x
Shorthand for the partial derivative of f with respect to x. Subscript order in higher partials reads left to right in the order the derivatives are taken.
Geometric meaning of a partial derivative
The slope of the trace obtained by slicing the graph with a plane holding the other variable constant, so it is a one-variable slope in disguise.
Second partial derivative
The result of differentiating a first partial again, with respect to the same or a different variable. A function of two variables has four second partials in total.
Mixed partial derivative
A second partial taken with respect to two different variables, such as differentiating first in x and then in y.
Clairaut theorem
If both mixed second partials are continuous on an open disk around a point, they are equal there. Equality can fail when that continuity hypothesis is dropped.
Tangent plane
The plane through a surface point whose normal is built from the two first partials, providing the best linear approximation to the graph near that point.
Linearisation
The affine function whose graph is the tangent plane, used to estimate values near a base point. Its accuracy degrades as the distance from the base point grows.
Differentiability in several variables
Requires the increment of the function to be captured by the linear part with an error vanishing faster than the distance moved. Existence of both partials alone is not enough.
Sufficient condition for differentiability
If both partial derivatives exist and are continuous on an open region, the function is differentiable there. This is the practical test used in almost every problem.
Total differential
The expression combining each partial derivative with the corresponding small change in its variable, used to estimate propagated error in measured quantities.
Chain rule with one parameter
For a function of variables that each depend on a single parameter, the derivative sums each partial times the derivative of its variable with respect to that parameter.
Chain rule with two parameters
For inner variables depending on two parameters, each output partial is a sum over the intermediate variables of a partial times a partial.
Tree diagram for the chain rule
A bookkeeping device drawing one branch per dependency path; multiply along each branch and add across branches to avoid dropping a term.
Implicit differentiation in several variables
Differentiating a relation defining one variable implicitly and solving for the wanted derivative, giving a ratio of partial derivatives valid where the denominator partial is nonzero.
Gradient vector
The vector assembled from all first partial derivatives of a scalar function, written grad f. It lives in the domain, not on the graph.
Directional derivative
The rate of change of a function in the direction of a specified unit vector, equal to the dot product of the gradient with that unit vector when the function is differentiable.
Unit vector requirement
The direction supplied to a directional derivative must be normalised; using an unnormalised vector inflates the answer by the vector's length, a very common error.
Gradient as direction of steepest ascent
At any point the gradient aims in the direction of maximum increase, and its magnitude is exactly that maximum rate of change.
Direction of steepest descent
The negative of the gradient, along which the function decreases fastest, at a rate equal in magnitude to the gradient length.
Gradient orthogonal to level sets
At every point the gradient is perpendicular to the level curve or level surface through that point, which is why it supplies normal vectors for tangent planes.
Zero directional derivative direction
Any direction orthogonal to the gradient produces no first-order change, which is exactly the tangent direction to the level set.
Tangent plane to a level surface
The plane through a point with the gradient of the defining function as its normal vector, valid wherever that gradient is nonzero.
Normal line to a surface
The line through a surface point in the direction of the gradient, perpendicular to the tangent plane there.
Partials exist yet function is discontinuous
A genuine possibility in several variables: both partials can exist at a point where the function is not even continuous, because partials only probe two directions.