Partial Derivatives

In Maxima there is no distinction between derivatives and partial derivatives. We can calculate

$\displaystyle \dfrac{\partial^{n+m}}{\partial x^n \partial y^n}f(x,y)$

--> diff(f(x,y),x,n,y,m)

Here we can define the function

and call it f(x,y) by using the function definition operator ":=". Then differentiate it, and evaluate the derivative at a particular value, for example,

--> f(x,y):= x^3 + x^2*y^3 - 2*y^2
--> f_xy: diff(f(x,y), x, 1, y, 1)
--> f_xy, x=2, y=1

The differential

can be obtained by the same operation "diff" when the variables are not specified

--> diff(f(x,y))

It returns the following output

   (3*x^2*y^2-4*y)*del(y)+(2*x*y^3+3*x^2)*del(x)

where "del(x)" and "del(y)" respectively represent the differential

and

Chain rule. When and in are the functions and of , Maxima applies the chain rule for the partial derivative and . Here we have to declare that and are dependent on the pair of variables.

--> f(x,y) := exp(x)*sin(y)
--> depends([x,y],[s,t])
--> f_s: diff(f(x,y), s)
--> f_t: diff(f(x,y), t)

In order to complete the calculation we have to substitute $\partial x/\partial s$ , $\partial y/\partial s$ , $\partial x/\partial t$ , $\partial y/\partial t$ by the respective results of partial derivative. Consider

and

we can complete the partial derivative via chain rule.

--> f_s, diff(x,s)=diff(s*t^2,s), diff(y,s)=diff(s^2*t,s)
--> f_t, diff(x,t)=diff(s*t^2,t), diff(y,t)=diff(s^2*t,t)

However, Maxima can compute the derivative by brute force when we construct the function

at the beginning.

--> z: f(x,y), x=s*t^2, y=s^2*t
--> diff(z, s)
--> diff(z, t)

You may use wxMaxima and complete some of the even-numbered exercises assigned in the class.

exercise14-3.wxm (the first and the second partial derivatives)
exercise14-4.wxm (tangent planes and differentials)
exercise14-5.wxm (chain rule, implicit differentiation)