## 2.11.3 Differentiation using the Reference Element

To find the derivative of \(\tilde{T}\) with respect to \(x\) (or similarly \(y\)) within an element, we differentiate the three nodal basis functions within the element:

To find the \(x\)-derivatives of each of the \(\phi _ j\)'s, the chain rule is applied:

Similarly, to find the derivatives with respect to \(y\):

The calculation of the derivatives of \(\phi _ j\) with respect to the \(\xi\) variables gives:

The only remaining terms are the calculation of \(\frac{\partial \xi _1}{\partial x}\), \(\frac{\partial \xi _2}{\partial x}\), etc. which can be found by differentiating Equation (2.270),

where

Note that the Jacobian, \(J\), is equal to twice the area of the triangular element.