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=Partial Derivatives=

=Partial Derivatives=

Partial derivatives are, most simply, a way to evaluate a derivative of a multivariable function, "partial" to only a single variable.  This has geometric meaning, which we will discuss in the next lecture.  The calculation of a partial derivative is the essence of simplicity - we simply treat the variables we do NOT wish to differentiate with respect to as constants.

::::Main article: [[Derivative#Partial_derivatives]].

For example, if we have a function <math>f(x,y) = x^2 y^3 \ </math>, we can write the partial derivatives as <math>\frac{\partial}{\partial x } f(x,y)= \partial_x f(x,y) = f_x (x,y)= 2xy^3 \ </math> and <math>f_y (x,y) = 3x^2 y^2 \ </math>.

When dealing with functions of more than one variable, one has to be careful about what a "derivative" is.  The operative concept is the ''partial derivative'', which is the derivative of the function with respect to a given argument ''while holding all the other arguments constant''.  It is written like a derivative, but with a different "d" character.

 

:<math>f(x,y,z) = \frac{x \sin{x} y^2}{z}\,</math>

 

we have

:<math>\frac{\partial}{\partial x} f(x,y,z) = \frac{y^2 (x \cos{x} + \sin{x})}{z}</math>

 

:<math>\frac{\partial}{\partial y} f(x,y,z) = \frac{2 y x \sin{x}}{z}</math>

 

:<math>\frac{\partial}{\partial z} f(x,y,z) = \frac{- x y^2 \sin{x}}{z^2}</math>

 

:<math>\frac{\partial}{\partial x} f(x,y,z) = \partial_x f(x,y,z) = f_x (x,y,z)</math>

=Problems=

=Problems=