# [SOLVED] Differentiation

• Feb 5th 2009, 11:18 AM
ronaldo_07
[SOLVED] Differentiation
u(x,y) = cos(4xy)e^(2x² - 2y² + 2)

v(x,y) = sin(4xy)e^(2x² - 2y² + 2)

Compute the partial derivatives of u and v with respect to x and y for all values (x, y) and verify that they satisfy the Cauchy-Riemann equations.

How do I differentiate each?

for example i got du/dx=4e^(2x² - 2y² + 2)cos(4xy)
du/dy=4ye^(2x² - 2y² + 2)cos(4xy)
dv/dx=4e^(2x² - 2y² + 2)sin(4xy)
dv/dy=4ye^(2x² - 2y² + 2)sin(4xy)

Im not sure if this is correct
• Feb 5th 2009, 11:57 AM
Aryth
You're almost right, for each partial derivative you MUST use the product rule, for example:

$\displaystyle \frac{\partial{u}}{\partial{x}} = \frac{\partial{[\cos{(4xy)}]}}{\partial{x}}e^{(2x^2 - 2y^2 - 2)} + \cos{(4xy)}\frac{\partial{[e^{(2x^2 - 2y^2 - 2)}}]}{\partial{x}}$

$\displaystyle = -4\sin{(4xy)}e^{(2x^2 - 2y^2 - 2)} + 4x\cos{(4xy)}e^{(2x^2 - 2y^2 - 2)}$

$\displaystyle = 4e^{(2x^2 - 2y^2 - 2)}[x\cos{(4xy)} - \sin{(4xy)}]$

You have to differentiate them as such to arrive at answers that you can use for the CR equations.