The concept of double rational Chebyshev functions on the semi-infinite domain () and some of their properties are introduced in this work. Also, the definition of derivatives for double rational Chebyshev functions is improved.  This new definition is employed to deal with partial differential equations with variable coefficients derived on the interval. The new definition with the spectral collocation method generates a new improved scheme. Numerical results are show that demonstrates the validity and applicability of the two techniques. The obtained numerical results are compared with the exact solution where it shown to be very attractive with good accuracy.