shape functions partial derivative

Hello !

I need some help to figure out a calculus problem. I must perform a partial derivative of a polynomial, p(x,y).

But to build this polynomial I use the shape function concepts (of Finite Element Method) and I get lost when I need to derivate it.

The whole formulation is on the picture file attached. The polynomial I need (equation 1) is obtained with equation 8, using equations 2 to 7. Note that I use exact formulaes (5b) to do the integrations on 5a.

I already obtain correct values for equation 8, given x and y coordinates. Next I do a coordinate transformation on equation 9. There my polynomial is represented by equation 10. Next I need to perform the derivatives of it, like in equations 11 and 12.

I tried the product rule of derivation for equation 7, using the derivative of an inverse matrix as -R^-1*(dR/dx)*R^-1 but didn't get any result.

I dont know if I was able to explain my problem very clear to you all, but any help will be appretiated.

Again, the whole idea is to compute the gradient of a property using the polynomial obtained with the shape functions.

Thanks in advance,

Carlos.