Partial derivative
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point.The partial derivative of a function f with respect to the variable x is represented as fx or <math>\frac{ \partial f }{ \partial x }</math> (where <math>\partial</math> is a rounded 'd' known as the 'partial derivative symbol').
As an example, consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula
- <math>V = \frac{ r^2 h \pi }{3}</math>
The partial derivative of V with respect to r is
- <math>\frac{ \partial V}{\partial r} = \frac{ 2r h \pi }{3}</math>
it describes the rate with which a cone's volume changes if its radius is increased and its height is kept constant. The partial with respect to h is
- <math>\frac{ \partial V}{\partial h} = \frac{ r^2 \pi }{3}</math>
and represents the rate with which the volume changes if its height is increased and its radius is kept constant.
Equations involving an unknown function's partial derivatives are called partial differential equations and are ubiquitous throughout science.
Formal definition and properties
Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Rn and f : U -> R a function. We define the partial derivative of f at the point a=(a1,...,an)∈U with respect to the i-th variable xi as
- <math>\frac{ \partial }{\partial x_i }f(a) =
Even if all partial derivatives ∂f/∂xi(a) exists at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that f is a C1 function.
The partial derivative ∂f/∂xi can be seen as another function defined on U and can again be partially differentiated. If all mixed partial derivatives exist and are continuous, we call f a C2 function; in this case, the partial derivatives can be exchanged:
- <math>\frac{\partial^2f}{\partial x_i} \partial x_j = \frac{\partial^2f} {\partial x_j} \partial x_i</math>
The vector consisting of all partial derivatives of f at a given point a is called the gradient of f at a:
- <math>\operatorname{grad}f(a) = \left( \frac{\partial f}{\partial x_1}(a), \dots , \frac{\partial f}{\partial x_n}(a) \right)</math>
If f is a C1 function, then grad f(a) has a geometrical interpretation: it is the direction in which f grows the fastest, the direction of steepest ascent.
But this it is, our foot
Shall stay with us- Order for sea.html">sea is given;
Where their appointment we may best.html">best discover
SCENE XI.
Which, as I take't, we shall; for his best force
And hold our best advantage. Exeunt
ACT_4|SC_12
A hill near Alexandria
Enter ANTONY and SCARUS
ANTONY. Yet they are not join'd. Where yond pine.html">pine does stand
Straight how 'tis like to go. Exit
In Cleopatra's sails their nests. The augurers
And dare not speak their knowledge. Antony
His fretted fortunes give him hope and fear
[Alarum afar off, as at a sea-fight]
Re-enter ANTONY
ANTONY. All is lost!
My fleet hath yielded to the foe, and yonder
Like friends long lost. Triple-turn'd whore! 'tis thou
Makes only wars on thee. Bid them all fly.html">fly;
I have done all. Bid them all fly; begone. Exit SCARUS
Fortune and Antony part here; even here
That spaniel'd me at heels, to whom I gave
On blossoming Caesar; and this pine is bark'd
O this false soul of Egypt! this grave charm-
Whose bosom was my crownet, my chief end-
Beguil'd me to the very heart of loss.
CLEOPATRA. Why is my lord enrag'd against his love?
And blemish Caesar's triumph. Let him take thee
Follow his chariot, like the greatest spot
For poor'st diminutives, for doits, and let
With her prepared nails. Exit.
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