SOLUTION: Differentiate the following expression implicitly: (1+e^3x)^2 = 3+ln(x+y)

Algebra ->  Customizable Word Problem Solvers  -> Misc -> SOLUTION: Differentiate the following expression implicitly: (1+e^3x)^2 = 3+ln(x+y)      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1155753: Differentiate the following expression implicitly: (1+e^3x)^2 = 3+ln(x+y)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Fortunately, you will never see an equation like this anywhere except in a math class, as an exercise in various concepts of calculus.

Find the derivative using implicit differentiation, using power and chain rules, among other things.

%281%2Be%5E%283x%29%29%5E2+=+3%2Bln%28x%2By%29

2%281%2Be%5E%283x%29%29%283e%5E%283x%29%29+=+0+%2B+%281%2F%28x%2By%29%29%281%2Bdy%2Fdx%29

%28x%2By%29%282%281%2Be%5E%283x%29%29%283e%5E%283x%29%29%29+=+1%2Bdy%2Fdx

dy%2Fdx+=+%28x%2By%29%282%281%2Be%5E%283x%29%29%283e%5E%283x%29%29%29-1 [1]

We can't leave the "y" in that expression, so we need to solve the original equation for y.

%281%2Be%5E%283x%29%29%5E2+=+3%2Bln%28x%2By%29
%281%2Be%5E%283x%29%29%5E2-3+=+ln%28x%2By%29
e%5E%28%281%2Be%5E%283x%29%29%5E2-3%29+=+x%2By
y+=+e%5E%28%281%2Be%5E%283x%29%29%5E2-3%29-x [2]

Now substitute [2] in [1].





Note as a further exercise in this kind of thing, you can use equation [2] above and differentiate it directly; that of course will give you the same answer.