Understanding Derivative of $\Vert f\Vert_{2}^{2}$

[cheeez]

$$\frac{d}{dp}(\Vert f\Vert_{p}^{2})\mid_{p=2}=\frac{d}{dp}((\int_{\mathbb{\mathbb{R}}^{n}}\mid f\mid^{p}dx)^{\frac{2}{p}})\ldots=\frac{1}{2}\int_{\mathbb{\mathbb{R}}^{n}}\mid f(x)\mid^{2}ln\left(\frac{\mid f(x)\mid^{2}}{\Vert f\Vert_{2}^{2}}\right)$$

this is the derivative evaluated at p=2.

does anyone see how this works? I started with the e^ln(whole integral) and then chain rule and so on...then plugged in p=2 at the end but couldn't get it.wow i cannot get the d/dp to show up properly something is wrong with the latex...

 

[Char. Limit]

$$\frac{d}{dp}(\Vert f\Vert_{p}^{2})\mid_{p=2}=\frac{d}{dp}((\int_{\mathbb{\mathbb{R}}^{n}}\mid f\mid^{p}dx)^{\frac{2}{p}})\ldots=\frac{1}{2}\int_{\mathbb{\mathbb{R}}^{n}}\mid f(x)\mid^{2}ln\left(\frac{\mid f(x)\mid^{2}}{\Vert f\Vert_{2}^{2}}\right)$$

this is the derivative evaluated at p=2.

does anyone see how this works? I started with the e^ln(whole integral) and then chain rule and so on...then plugged in p=2 at the end but couldn't get it.





wow i cannot get the d/dp to show up properly something is wrong with the latex...

Did that fix it?