Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor

Question

I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y).

Eventually I managed to pull the (presumably) correct function: $$f(x,y) = \ln \left( (1+2x)^{\frac{5}{4}y} - (1-x)^y - 1 \right)$$ around $\left( 0, 0.2 \right)$.

But its partial derivatives are overly complicated, see examples for $f_{xx}$ and $f_{yx}$.
So my best guess for the function is wrong. It feels like I'm missing an identity that would simplify the task.

Could you please direct my towards the most appropriate function?

Answer 1

My opinion is that you complicate things too much. Just try $f(x,y)=\ln (x^{0.25} + y^{0.02} -1)$ around $(1,1)$.