SOLUTION: Please help. I couldn't find any likes of a solution to this problem nor a section to enter this problem on the website : Given that {{{y=2^x*x^2}}}, prove that {{{(dy)/(dx) =(y

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please help. I couldn't find any likes of a solution to this problem nor a section to enter this problem on the website : Given that {{{y=2^x*x^2}}}, prove that {{{(dy)/(dx) =(y      Log On


   

Question 111639: Please help. I couldn't find any likes of a solution to this problem nor a section to enter this problem on the website :
Given that y=2%5Ex%2Ax%5E2, prove that %28dy%29%2F%28dx%29+=%28y%282+%2B+x%2Aln%282%29%29%29%2Fx.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Please help. I couldn't find any likes of a solution to this problem nor a section to enter this problem on the website :
Given that y=2%5Ex%2Ax%5E2, prove that %28dy%29%2F%28dx%29+=%28y%282+%2B+x%2Aln%282%29%29%29%2Fx.
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y=2%5Ex%2Ax%5E2

take the 1n of both sides:

ln%28y%29=ln%282%5Ex%2Ax%5E2%29

ln%28y%29=ln%282%5Ex%29+%2Bln%28x%5E2%29

ln%28y%29=x%2Aln%282%29+%2B2%2Aln%28x%29

Now, upon realizing that ln%282%29 is just a constant,
we can take the derivative of both sides:

%28%28dy%29%2F%28dx%29%29%2Fy=ln%282%29+%2B+2%2A%281%2Fx%29

or 

%28%28dy%29%2F%28dx%29%29%2Fy=ln%282%29+%2B+2%2Fx

Clear of fractions by multiplying both sides by xy:

xy%2A%28%28dy%29%2F%28dx%29%29%2Fy+=xy%2Aln%282%29+%2B+xy%2A%282%2Fx%29

Cancel the y's on the left, and cancel the x's in the
rightmost term:

x%2A%28%28dy%29%2F%28dx%29%29+=xy%2Aln%282%29+%2B+2y

Factor out y on the right:

x%2A%28%28dy%29%2F%28dx%29%29+=y%28x%2Aln%282%29+%2B+2%29

Divide both sides by x

%28dy%29%2F%28dx%29+=%28y%28x%2Aln%282%29+%2B+2%29%29%2Fx

Reverse the two terms in the parentheses:

%28dy%29%2F%28dx%29+=%28y%282+%2B+x%2Aln%282%29%29%29%2Fx

Edwin