# Jacobian when there's a multivariate function inside it

## Homework Statement

differentiate the function F(x,y) = f( g(x)k(y) ; g(x)+h(y) )

## Homework Equations

Standard rules for partial differentiation

## The Attempt at a Solution

The Jacobian will have two columns because of the variables x and y. But what then? f is a multivariate function inside the Jacobian!

BvU
Homework Helper
So on top of the standard rules you get the chain rule.
Show some attempt at solution and help is on the way.

To demo my ignorance: Differentiating gives two columns, but one row only, right ?
Is there a significance in the ";" ? You write F ( x , y ) -- a notation which I am also familiar with -- , but then you write f ( u ; v )

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

differentiate the function F(x,y) = f( g(x)k(y) ; g(x)+h(y) )

## Homework Equations

Standard rules for partial differentiation

## The Attempt at a Solution

The Jacobian will have two columns because of the variables x and y. But what then? f is a multivariate function inside the Jacobian!

Do you mean ##F(x,y) = f(u,v)##, where ##u = g(x) k(y)## and ##v = g(x) + h(y)##? If so, just apply the chain rule for derivatives. You need to express the answers in terms of the functions ##f_1, f_2##, where ##f_1(u,v) \equiv \partial f(u,v)/\partial u## and ##f_2(u,v) \equiv \partial f(u,v) / \partial v##.

Consider the partial derivatives that make up the derivative matrix. It should be a 2x2, you have two functions, and take the derivative of both functions wrt x or wrt y.