Example
Alice's utility function is `u \left( X, Y \right) = XY`.
Her marginal rate of substitution is the ratio of the marginal utilities `MU_X` and `MU_Y`: $$ MRS = - \frac{MU_X}{MU_Y} = - \frac{Y}{X} $$
Question
Alice's utility function is `u (X, Y) = X^{340} Y^{759}`.
What is her Marginal Rate of Substitution?
$$ MU_X = \frac{du(X, Y)}{dX} = 340 X^{340 - 1} Y^{759} = 340 X^{339} Y^{759} $$
$$ MU_Y = \frac{du(X,Y)}{dY} = 759 X^{340} Y^{759 - 1} = 759 X^{340} Y^{758} $$
Therefore $$ MRS = - \frac{MU_X}{MU_Y} = - \frac{340 X^{339} Y^{759}}{759 X^{340} Y^{758}} = - \frac{340 Y}{759 X} $$