Use logarithmic differentiation to find the derivative of y with respect to the independent variable:

y = (x+2)^x

Question

Use logarithmic differentiation to find the derivative of y with respect to the independent variable: 

y = (x+2)^x

psymon · Answer

Well, you would first start with taking the natural log of both sides. You would have 
ln(y) = ln[(x+2)^x], which would then become ln(y) = xln(x+2). Have you made it that far?

anonymous · Answer

Yes, I'm just not sure how to get the derivative of a natural logarithm. I thought it would be x/(x+2), but that would just cause it to equal zero.

psymon · Answer

Well, its u'/u. But the derivative of x+2 is simply 1. Constants disappear and x to thefirst power just drops and leaves behind its coefficient, which is one in this case. So you would have a product rule, knowing that thederivative of ln(x+2) is just 1/(x+2).

anonymous · Answer

So, using the product rule, it would be x * (1/(x + 2)) + (xln(x + 2)) * 1, which comes out to x/(x + 2) + xln(x+2)?

psymon · Answer

Except you have an extra x with the xln(x+2) It should just be ln(x+2) for that part.

anonymous · Answer

Ah okay! Thank you very much! :D

psymon · Answer

You know where to go from there?

anonymous · Answer

This can be simplified? (Sorry, I'm REALLY bad with logarithms for some reason.)

psymon · Answer

Well, you would have:

psymon · Answer

|dw:1375134313374:dw|

From here, you need to multiply both sides by y to solve for dy/dx. After you get dy/dx by itself, you have to substitute the original equation in for y.