# Chain rule and disagreement with the book's answer.

• Sep 24th 2010, 03:41 PM
artabro
Chain rule and disagreement with the book's answer.
The exercise states: "Find dz/dt using the chain rule"
F(x,y) = xLny + yLnx, x=t+1 y=Lnt

(Lny+(x/y)) + (Lnx + (y/x))*(1/t)

(Lny+(y/x)) + (Lnx + (x/y))*(1/t)

I don´t understand why they inverse the x/y division. Could any one help me please? :S
• Sep 24th 2010, 04:03 PM
Ulysses
$f_x=\ln y+\dysplaystyle\frac{y}{x}$

$f_y=\dysplaystyle\frac{x}{y}+\ln x$

$x_t=1$

$y_t=\dysplaystyle\frac{1}{t}$

Then the chain rule gives:
$(\ln y+\dysplaystyle\frac{y}{x},\dysplaystyle\frac{x}{y }+\ln x)*(1,\dysplaystyle\frac{1}{t})=$ (*) is the dot product.

$=\ln y+\dysplaystyle\frac{y}{x}+(\dysplaystyle\frac{x}{ y}+\ln x)(\dysplaystyle\frac{1}{t})$