\(\displaystyle \L\\x(x^{2}+4)^{\frac{1}{2}}\)
Product rule:
\(\displaystyle (x)(\frac{1}{2})(x^{2}+4)^{\frac{-1}{2}}(2x)+(x^2+4)^{\frac{1}{2}}\)
You are correct.
Simplify:
\(\displaystyle \L\\\sqrt{x^{2}+4}+\frac{x^{2}}{\sqrt{x^{2}+4}}\)
Multiply the left side, top and bottom, by \(\displaystyle \sqrt{x^{2}+4}\)
\(\displaystyle \L\\\frac{\sqrt{x^{2}+4}}{\sqrt{x^{2}+4}}\frac{\sqrt{x^{2}+4}}{1}+\frac{x^{2}}{\sqrt{x^{2}+4}}\)
=\(\displaystyle \L\\\frac{2x^{2}+4}{\sqrt{x^{2}+4}}\)
Now, take the derivative of that?. You can use the product rule as well.
Just use \(\displaystyle \L\\(2x^{2}+4)(x^{2}+4)^{\frac{-1}{2}}\)
Give it a shot and let us know what you came up with.
