6)
a) let f(x,y) = 4x^2 - 6xy + 4y^2 - 16, then compute df/dx and df/dy. then dy/dx = (df/dx)/(df/dy)
b) assuming you mean the value of dy/dx, just plug in the values to the expression found in a). i get -1 for both.
c) the graph of the equation has derivative -1 at the aforementioned points, ie the tangent line makes a 45 degree angle with the horizontal axis.
7 and 8 were answered, hope that's enough help :)
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http://speeddemosarchive.com/MetroidPrime.html#PAL100