[tex]y=\sqrt x\,e^{x^2}(x^2+5)^{12}=x^{1/2}e^{x^2}(x^2+5)^{1/2}[/tex]
Take the logarithm of both sides and expand the right hand side:
[tex]\ln y=\ln\left(x^{1/2}e^{x^2}(x^2+5)^{1/2}\right)[/tex]
[tex]\ln y=\ln x^{1/2}+\ln e^{x^2}+\ln(x^2+5)^{12}[/tex]
[tex]\ln y=\dfrac12\ln x+x^2\ln e+12\ln(x^2+5)[/tex]
[tex]\ln y=\dfrac12\ln x+x^2+12\ln(x^2+5)[/tex]
Now take the derivative of both sides with respect to [tex]x[/tex]:
[tex]\dfrac1y\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1{2x}+2x+\dfrac{24x}{x^2+5}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\left(\dfrac1{2x}+2x+\dfrac{24x}{x^2+5}\right)y[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\left(\dfrac1{2x}+2x+\dfrac{24x}{x^2+5}\right)\sqrt x\,e^{x^2}(x^2+5)^{12}[/tex]
I'd stop there, but you could condense the right side a bit to get
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4x^4+21x^2+48x+5}{2x(x^2+5)}\sqrt x\,e^{x^2}(x^2+5)^{12}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4x^4+21x^2+48x+5}{2\sqrt x}e^{x^2}(x^2+5)^{11}[/tex]
