MHB Solving a Tricky Chain Rule Question with Confusing Variables

[Yankel]

Hello,

I have a tricky chain rule question, I think understanding it is more difficult than solving.

For the function z=f(x,y) it is given that:

f_{y}(0,-3)=-2

and

\[f_{x}(0,-3)=3\]

so for the function

\[g(x,y)=f(2\cdot ln(x+y),x^{4}-3y^{2})\]

choose the correct answer:

(1)
\[g_{y}(0,-3)=18\]

(2)
\[g_{y}(0,-3)=-6\]

(3)
\[g_{y}(0,1)=18\]

(4)
\[g_{y}(0,1)=-6\]

(5)
Non of the above answers

I am confused slightly. I thought to call

\[u=2\cdot ln(x+y)\]
and
\[v=x^{4}-3y^{2}\]

but when I put x=0 and y=-3, I can't get a value for u, since I get a negative value under the ln.

Can you assist clearing this up ? Thank you !

 

[Prove It]

Their writing of the function is a bit sloppy. Remember $\displaystyle \begin{align*} 2\ln{(x+y)} = \ln{ \left[ (x + y)^2 \right] } \end{align*}$, which means you CAN have values of x and y such that $\displaystyle \begin{align*} x + y < 0 \end{align*}$, as squaring will make them positive.

 

[Yankel]

I used the chain rule, and I called u= 2ln(x+y) and v=x^4-3y^2

I don't know how to proceed from here.