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^{475} Y^{92}`.
What is her Marginal Rate of Substitution?
$$ MU_X = \frac{du(X, Y)}{dX} = 475 X^{475 - 1} Y^{92} = 475 X^{474} Y^{92} $$
$$ MU_Y = \frac{du(X,Y)}{dY} = 92 X^{475} Y^{92 - 1} = 92 X^{475} Y^{91} $$
Therefore $$ MRS = - \frac{MU_X}{MU_Y} = - \frac{475 X^{474} Y^{92}}{92 X^{475} Y^{91}} = - \frac{475 Y}{92 X} $$