1. ## derivative

1) $\displaystyle z=\frac{1}{\sqrt{x^2+4y^2}}$

a) $\displaystyle \frac{\partial^2 z}{\partial x^2}$

b) $\displaystyle \frac{\partial^2 z}{\partial x \partial y}$

a) $\displaystyle -\frac{\sqrt{x^2+4y^2}+2x^2}{x^2+4y^2}$

b) $\displaystyle -\frac{4xy}{\sqrt{x^2+4y^2}}$

2)
$\displaystyle z=ln(x^2+y^2)$
a) $\displaystyle \frac{\partial^3 z}{\partial x \partial y^2}$

a) $\displaystyle \frac{4x^2+4y^2-8x^3-8xy^2}{(x^2+y^2)^3}$

3) $\displaystyle z=\sqrt{2xy+y}$
a) $\displaystyle \frac{ \partial^2 z}{ \partial x \partial y}$

b) $\displaystyle \frac{ \partial^3 y}{ \partial x^3}$

How I do the 3?

2. Originally Posted by Apprentice123
1) $\displaystyle z=\frac{1}{\sqrt{x^2+4y^2}}$

a) $\displaystyle \frac{\partial^2 z}{\partial x^2}$

b) $\displaystyle \frac{\partial^2 z}{\partial x \partial y}$

a) $\displaystyle -\frac{\sqrt{x^2+4y^2}+2x^2}{x^2+4y^2}$
nope

begin by writing the original as $\displaystyle z = (x^2 + 4y^2)^{-1/2}$

b) $\displaystyle -\frac{4xy}{\sqrt{x^2+4y^2}}$
nope

2)
$\displaystyle z=ln(x^2+y^2)$
a) $\displaystyle \frac{\partial^3 z}{\partial x \partial y^2}$

a) $\displaystyle \frac{4x^2+4y^2-8x^3-8xy^2}{(x^2+y^2)^3}$
nope

3) $\displaystyle z=\sqrt{2xy+y}$
a) $\displaystyle \frac{ \partial^2 z}{ \partial x \partial y}$

b) $\displaystyle \frac{ \partial^3 y}{ \partial x^3}$

How I do the 3?
first write as $\displaystyle z = (2xy + y)^{1/2}$

for (a), first differentiate with respect to x, then differentiate with respect to y

for (b), take $\displaystyle \frac {\partial z}{\partial x}$ and differentiate it with respect to x twice