y=3x³e²ˣ/ln(x)
Let u=3x³e²ˣ, du/dx=9x²e²ˣ+6x³e²ˣ=3x²e²ˣ(3+2x);
Let v=ln(x), dv/dx=1/x.
y=u/v, dy/dx=(v(du/dx)-u(dv/dx))/v²,
dy/dx=(3x²e²ˣ(3+2x)ln(x)-3x²e²ˣ)/ln²(x),
dy/dx=3x²e²ˣ[(3+2x)ln(x)-1]/ln²(x).
Let u=3x²e²ˣ[(3+2x)ln(x)-1],
du/dx=(6xe²ˣ+6x²e²ˣ)[(3+2x)ln(x)-1]+3x²e²ˣ[2ln(x)+(3+2x)/x],
du/dx=6xe²ˣ(1+x)[(3+2x)ln(x)-1]+6x²e²ˣln(x)+3xe²ˣ(3+2x),
du/dx=3xe²ˣ[2(1+x)[(3+2x)ln(x)-1]+2xln(x)+3+2x],
du/dx=3xe²ˣ[2(3+6x+2x²)ln(x)+1];
Let v=ln²(x), dv/dx=2ln(x)/x.
dy/dx=u/v,
d²y/dx²=(v(du/dx)-u(dv/dx))/v²,
d²y/dx²=3xe²ˣ[ln²(x)([2(3+6x+2x²)ln(x)+1]-[(3+2x)ln(x)-1](2ln(x))]/ln⁴(x),
d²y/dx²=3xe²ˣ{([2(3+6x+2x²)ln(x)+1]/ln²(x)-[2(3+2x)ln(x)-1]/ln³(x)}.