I am stuck at this partial derivative question. Someone earlier helped me with the first part of the question but i still cant understand the 2nd part of the question involving second order partial derivative.
here is the solution that a member helped me with earlier, i can't figure out how to get to the last line from the 3rd last line in the solution posted below. Some help with would be much appreciated.
Substitute $$\frac{\partial z}{\partial x}=\frac{-z}{z^2+x}$$ into the equation $$\frac{\partial^2 z}{\partial x^2}(z^2+x)=-2(z(\frac{\partial z}{\partial x})^2+\frac{\partial z}{\partial x})$$
Then isolate the $\frac{\partial^2 z}{\partial x^2}$ term by dividing both sides by $(z^2+x)$. You may need to nudge the fractions a bit for a common denominator so that it simplifies nicely.
Note the multiplication by $1=\frac{z^2+x}{z^2+x}$ in the second term for a common denominator:
$$\frac{\partial^2 z}{\partial x^2}(z^2+x)=-2(z(\frac{-z}{z^2+x})^2-\frac{z}{z^2+x})\\ =-2(\frac{z^3}{(z^2+x)^2}-\frac{z}{z^2+x}\cdot\frac{z^2+x}{z^2+x})\\ =-2(\frac{z^3}{(z^2+x)^2}-\frac{z^3+xz}{(z^2+x)^2})=\frac{2xz}{(z^2+x)^2} $$
Dividing both sides once more by $(z^2+x)$ yields the desired result.