# Determine if goods are substitutes or Complements based on demand function

So I have a consumer with a utility function of the Cobb-Douglas form $$v(x_1,x_2)=x^{\frac{1}{2}}_1x^{\frac{1}{2}}_2$$.

From that I constructed the demand function for good 1 and good 2:
$$x_1=\frac{1}{2} \cdot \frac{m}{p_1}$$

$$x_2=\frac{1}{2} \cdot \frac{m}{p_2}$$

From here, I am to determine whether good 1 and good 2 are substitutes or complements. I was told by my teacher to take the derivative with respect to $$m, p_1$$ and $$p_2$$, however I can't get a sensible result. As far as I've come to understand from my book, I need to find $$\frac{dx_1}{dp_2}$$, the change in the demand for good 1 as the price of good 2 changes. Substituting m back to $$p_1x_1+p_2x_2$$ becomes messy as well. I understand the problem theoretically, but I simply can't figure out how to properly argue using math, rather than logic.

I hope someone is willing to help

• Would be useful to tell us what you get for each step before it gets "messy." – Art Dec 11 '19 at 4:29