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^x*x^2}}}, prove that {{{(dy)/(dx) =(y(2 + x*ln(2)))/x}}}.
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{{{y=2^x*x^2}}}

take the {{{1n}}} of both sides:

{{{ln(y)=ln(2^x*x^2)}}}

{{{ln(y)=ln(2^x) +ln(x^2)}}}

{{{ln(y)=x*ln(2) +2*ln(x)}}}

Now, upon realizing that {{{ln(2)}}} is just a constant,
we can take the derivative of both sides:

{{{((dy)/(dx))/y=ln(2) + 2*(1/x)}}}

or 

{{{((dy)/(dx))/y=ln(2) + 2/x}}}

Clear of fractions by multiplying both sides by {{{xy}}}:

{{{xy*((dy)/(dx))/y =xy*ln(2) + xy*(2/x)}}}

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

{{{x*((dy)/(dx)) =xy*ln(2) + 2y}}}

Factor out y on the right:

{{{x*((dy)/(dx)) =y(x*ln(2) + 2)}}}

Divide both sides by {{{x}}}

{{{(dy)/(dx) =(y(x*ln(2) + 2))/x}}}

Reverse the two terms in the parentheses:

{{{(dy)/(dx) =(y(2 + x*ln(2)))/x}}}

Edwin</pre>