I am having trouble seeing how the chain rule gives the following result when used to solve a second order PDE.
Here are the two given characteristic equations
$E = y - x^2$ and $N = y$
We were also given the first derrivative with respect to x and asked to use the chain rule to find the second derrivative. I do not see how they got the answer they did.
$\Phi_x = -2x\Phi_E $
$\Phi_{xx}= ? $
My working gave me the follwing $\Phi_{xx} = (-2x\Phi_{EE})(-2x) = 4x^2\Phi_{EE}$
The given solution was $-2\Phi_E + 4x^2\Phi_{EE}$
Can someone explain how they got the additional term ?
Thank you