Changes

=Partial Derivatives=

=Partial Derivatives=

As in the single variable situation, we shall wish to study the rate of change of functions.  But how to do this?  In the single variable case, we had only one line along which to compute the slope, or rate of change, of a function.  But in the multivariate case, we have an infinite number of lines along which to compute the slope! 

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.

 

We proceed simply by discussing how to compute the rate of change only in the ''x'' and ''y'', and maybe ''z'', directions.  This is by something called '''partial derivatives.''' The concept 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.

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>.

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>.