# partial derivative question

• Feb 1st 2009, 08:10 PM
Frostking
partial derivative question
If g(x,y) = ln (ye^xy) g sub x = ? My answers show it is simply y but I do not understand why. I thought since the derv of a ln is 1/(ye^xy) and since y is constant in this case the answer should be 1/e^x?????????
• Feb 1st 2009, 08:16 PM
Chris L T521
Quote:

Originally Posted by Frostking
If g(x,y) = ln (ye^xy) g sub x = ? My answers show it is simply y but I do not understand why. I thought since the derv of a ln is 1/(ye^xy) and since y is constant in this case the answer should be 1/e^x?????????

Note that $\displaystyle g\!\left(x,y\right)=\ln\!\left(ye^{xy}\right)=\ln\ !\left(y\right) + \ln\!\left(e^{xy}\right)=\ln\!\left(y\right)+xy$

Now, when you partially differentiate with respect to x, y is treated as a constant.

Thus, $\displaystyle \frac{\partial g}{\partial x}=\frac{\partial}{\partial x}\left[\ln y\right]+y\frac{\partial}{\partial x}\left[x\right]=0+y=\color{red}\boxed{y}$

Does this make sense?
• Feb 1st 2009, 11:09 PM
Frostking