I'm going to solve for the partial derivatives via the Quotient Rule first, and solve it another, slightly easier way at the bottom of the page. 

First, we need to know what the Quotient Rule is:


or, a way I like to memorize this as: think of f as ‘high’ and g at ‘hoe’, say:

memorize: “hoe d-hi minus hi d-hoe, all over hoehoe

Taking q_A first;


Applying the Quotient Rule:

Simplifying this:
                                                the first part of our answer

Now turning to q_B,


Quotient Rule time:

                                        the second part of our answer.

Alternative Steps to the Solution 
You also don’t need to use the Quotient Rule at all for this example. You can turn it into a normal differentiation – you can do this since for each partial, we consider one price as constant (did you notice how the quotient rule was relatively easy with a zero in one of the top two terms?)

q_A first again:

                                                Same as above

Now for q_B:

Again, although written a little differently, it's the same as how we find it via the quotient rule. 


Complements, Substitute (Competitive Good) or Neither?

     p_A & p_B are both prices – we therefore assume that prices are positive and non-zero (think for a moment if prices were zero or negative... what do you think that would imply about quantity demanded? Thus prices are typically always assumed positive). 

That means that  are both negative. 

Showing that work:

    With both cross-partials (both cross-elasticities) negative, this implies the goods are complements. 

    With a negative cross-elasticity, as the price of one good goes up (alternatively; down), the quantity demanded for the other good goes decreases (alternatively; goes up). This is the definition of a complementary good.