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If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C 2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry. As with ordinary derivatives, multiple notations exist: the partial derivative of a function (,, …
A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. [1]: 26ff A partial derivative may be thought of as the directional derivative of the function along a coordinate axis. Partial derivatives may be combined in interesting ways to create more complicated ...
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h ( x ) = f ( x ) g ( x ) {\displaystyle h(x)={\frac {f(x)}{g(x)}}} , where both f and g are differentiable and g ( x ) ≠ 0. {\displaystyle g(x)\neq 0.}
The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions , which are smooth and certainly ...
For example, the second order partial derivatives of a scalar function of n variables can be organized into an n by n matrix, the Hessian matrix. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a ...
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R -derivation on the algebra of differentiable functions on a differentiable manifold ; more generally it is a derivation on the tensor algebra of a manifold.