Question

Use logarithmic differentiation to find the derivative of the function. y = sqrt(x)e^x^2(x^2 + 5)^12

Answer 1

`y=\sqrt x\,e^(x^2)(x^2+5)^(12)=x^(1/2)e^(x^2)(x^2+5)^(1/2)`

Take the logarithm of both sides and expand the right hand side:

`\ln y=\ln\left(x^(1/2)e^(x^2)(x^2+5)^(1/2)\right)`

`\ln y=\ln x^(1/2)+\ln e^(x^2)+\ln(x^2+5)^(12)`

`\ln y=\frac12\ln x+x^2\ln e+12\ln(x^2+5)`

`\ln y=\frac12\ln x+x^2+12\ln(x^2+5)`

Now take the derivative of both sides with respect to
`x`:

`\frac1y(\mathrm dy)/(\mathrm dx)=\frac1{2x}+2x+(24x)/(x^2+5)`

`(\mathrm dy)/(\mathrm dx)=\left(\frac1{2x}+2x+(24x)/(x^2+5)\right)y`

`(\mathrm dy)/(\mathrm dx)=\left(\frac1{2x}+2x+(24x)/(x^2+5)\right)\sqrt x\,e^(x^2)(x^2+5)^(12)`

I'd stop there, but you could condense the right side a bit to get

`(\mathrm dy)/(\mathrm dx)=(4x^4+21x^2+48x+5)/(2x(x^2+5))\sqrt x\,e^(x^2)(x^2+5)^(12)`

`(\mathrm dy)/(\mathrm dx)=(4x^4+21x^2+48x+5)/(2\sqrt x)e^(x^2)(x^2+5)^(11)`