Simplifing the 1st derivative

[meshun01]

1.) x((x^2+4)^(1/2)) I have to find the 1st and 2nd derivative.

I do not know how to simplify the first derivative.
Here is what I got by using the product rule.

((x^2+4)^(1/2))+((1/2)(x^2+4)^(-1/2))*(2x^2)

 

[galactus]

\(\displaystyle \L\\x(x^{2}+4)^{\frac{1}{2}}\)

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.

 

[meshun01]

So how would I find the critical number for the critical numbers