Question 153230
Solve equations for x:
<pre>
{{{15x^5 -20x^4 =6x^3 -8x^2}}}

{{{15x^5 -20x^4 -6x^3 +8x^2=0}}}

{{{x^2(15x^3 -20x^2 -6x +8)=0}}}

{{{x^2(5x^2(3x-4)-2(3x-4))=0}}}

{{{x^2((3x-4)(5x^2-2))=0}}}

{{{x^2(3x-4)(5x^2-2)=0}}}

{{{x^2=0}}}, {{{3x-4=0}}}, {{{5x^2-2=0}}}

  {{{x=0}}},  {{{x=4/3}}}, {{{5x^2=2}}}
                 {{{x^2=2/5}}}
                 {{{x}}}=±{{{sqrt(2/5)}}}                               
                 {{{x}}}=±{{{sqrt((2*5)/(5*5))}}}                 
                 {{{x}}}=±{{{sqrt(10/25)}}}
                 {{{x}}}=±{{{sqrt(10)/5}}}          
Determine the derivative of the following 
functions using positve exponents whenever 
necessary:

{{{f(x) = ln(3x^4 + 4x^3 + 6x^2)}}}

Use the formula:

{{{d/(dx)}}}{{{ln(u) = (du/dx)/u}}}

With {{{u=(3x^4 + 4x^3 + 6x^2)}}}
     {{{du/(dx)=12x^3+12x^2+12x}}}


{{{f}}}'{{{(x) = (12x^3 + 12x^2 + 12x)/(3x^4 + 4x^3 + 6x^2)}}}

{{{f}}}'{{{(x) = (12x(x^2 + x + 1))/(x^2(3x^2 + 4x + 6))}}}

{{{f}}}'{{{(x) = (12(x^2 + x + 1))/(x(3x^2 + 4x + 6))}}}

----------------

{{{f(x) = e^(2x)tan(x)}}}

Use these formulas:

{{{d(uv)/(dx)=u(dv/dx)+v(du/dx)}}}

{{{d(e^u)/(dx)= (e^u)(du/dx)}}}

{{{d(tan(u))/(dx)= (sec^2u)(du/dx)}}}

{{{u=e^(2x)}}}
{{{(du)/dx=(e^(2x))2}}}
{{{(du)/dx=2e^(2x)}}}

{{{v=tan(x)}}}
{{{(dv)/dx=(sec^2x)1}}}
{{{(dv)/dx=sec^2x}}}
{{{f(x) = e^(2x)tan(x)}}}
{{{f}}}'{{{(x) = e^(2x)sec^2x+tan(x)*2e^(2x)}}}
{{{f}}}'{{{(x) = e^(2x)sec^2x+2tan(x)e^(2x)}}}
{{{f}}}'{{{(x) = e^(2x)(sec^2x+2tan(x))}}}

Edwin</pre>