Question

`y = ( √(3x) )/(1 + e^(2x) )`
how would one differentiate this?
​

Answer 1

`~~~~~~~~y = (√(3x))/( 1+ e^(2x))\\\\\\\implies (dy)/(dx) = (d)/(dx) \left( (√(3x))/(1+ e^(2x)) \right)\\\\\\~~~~~~~~~~~~=((1 + e^(2x) ) (d)/(dx) \left(√(3x) \right) - \left(√(3x) \right)(d)/(dx)(1+e^(2x)))/(\left( 1+ e^(2x) \right)^2)~~~~~~~~~~~~;[\text{Quotient rule}]\\\\\\~~~~~~~~~~~~=((1+e^(2x)) \cdot (1)/(2√(3x)) \cdot 3-\left(√(3x) \right) \cdot 2 e^(2x))/((1 + e^(2x))^2)~~~~~~~~~~~~~~~~~~~~;[\text{Chain rule}]\\\\\\`

`=((\sqrt 3 (1 + e^(2x)))/(2\sqrt x) -2e^(2x) √(3x))/((1+e^(2x))^2)\\\\\\=\frac{\tfrac{\sqrt 3(1 +e^(2x)) - 4x\sqrt 3 e^(2x)}{2\sqrt x}}{(1+e^(2x))^2}\\\\\\=(\sqrt 3(1+e^(2x) -4xe^(2x)))/(2\sqrt x(1 +e^(2x))^2)`