Multidimensional chain rule, online calculator

Question

I would like to check my solutions for the derivative with the multidimensional chain rule, and I would like to do that online with a calculator. But I can not find one and I do not know how to use wolframalpha for it, which I am sure that it is capable of doing those.

I am talking about tasks like this for example:

$f:\mathbb{R}^2\to\mathbb{R}$, $f(u,v)=u^2+v^2$, $g:\mathbb{R}\to\mathbb{R}^2$, $g(t)=(e^t, t^2)$

And then derive $D(f\circ g)(t)$.

Do you know an online program, which can I use to check my solution, or how I can use wolframalpha for it?

Thanks in advance.

Answer 1

I'm not sure this is exactly what you want, but if not, it should at least give you ideas what to try:

d/dt ReplaceAll[u^2+v^2,{u->e^t,v->t^2}]

Answer 2

You can use the free Wolfram Cloud Sandbox to test these.

D[ u^2 + v^2 /. {u -> Exp[t], v -> t^2}, t]

or

D[ Function[{u, v}, u^2 + v^2] @@
   Function[t, {Exp[t], t^2}] @ t, t]

or some other possibilites that the Wolfram Language allows.

For another example, try this

Dt[u v + v w - u w /. {u -> x + y, v -> x + y^2,
   w -> x^2 + y}] // Simplify

or

Dt[Function[{u, v, w}, u v + v w - u w] @@ 
   Function[{x, y}, {x + y, x + y^2, x^2 + y}] @@
   {x, y}] // Simplify

or, again, some other possibilities. If you want a matrix try

{D[#, x], D[#, y]} & @ (u v + v w - u w /.
 {u -> x + y, v -> x + y^2,  w -> x^2 + y}) // Simplify