Question 163503
Differentiate with respect to x, 
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a) {{{y=(2x)/(e^x)}}}

Write with constant coefficient {{{2}}} outside
   {{{y=2(x/e^x)}}}

Write denominator as a factor of numerator with negative
exponent:

   {{{y=2(x*e^(-x))}}}

Use product rules for derivative {{{d(u*v)/dx=u(dv/dx)+v(du/dx)}}}
and {{{d(e^u)/dx=(e^u)(du/dx)}}}

of the inside of parentheses:

   {{{dy/dx=2(x*e^(-x)*(-1)+e^(-x)*(1))}}}  
   {{{dy/dx=2(-x*e^(-x)+e^(-x))}}} 
factor out {{{e^(-x)}}}
   {{{dy/dx=2(e^(-x)(-x+1))}}}
   {{{dy/dx=2(e^(-x)(1-x))}}}
Write {{{e^(-x)}}} in the numerator as {{{e^x}}} in the denominator
   {{{dy/dx=(2(1-x))/(e^x)}}}

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b) {{{y=3x*sin(2x)}}}

Write with constant coefficient {{{3}}} outside

   {{{y=3(x*sin(2x))}}}

Use product rule for derivative {{{d(u*v)/dx=u(dv/dx)+v(du/dx)}}}
and {{{d(sin(u))/dx=cos(u)(du/dx)}}}

of the inside of parentheses:

  {{{dy/dx=3(x*cos(2x)*(2)+sin(2x)*(1))}}} 

  {{{dy/dx=3(2x*cos(2x)+sin(2x))}}}

Edwin</pre>