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`:
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{Y}{X} $$Question
Alice's utility function is `u (X, Y) = X^{684} Y^{689}`.
What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?
$$
MU_X = \frac{du(X, Y)}{dX} = 684 X^{684 - 1} Y^{689} = 684 X^{683} Y^{689}
$$
$$
MU_Y = \frac{du(X,Y)}{dY} = 689 X^{684} Y^{689 - 1} = 689 X^{684} Y^{688}
$$
Therefore
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{684 X^{683} Y^{689}}{689 X^{684} Y^{688}} = - \frac{684 Y}{689 X} $$