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^{569} Y^{848}`.
What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?
$$
MU_X = \frac{du(X, Y)}{dX} = 569 X^{569 - 1} Y^{848} = 569 X^{568} Y^{848}
$$
$$
MU_Y = \frac{du(X,Y)}{dY} = 848 X^{569} Y^{848 - 1} = 848 X^{569} Y^{847}
$$
Therefore
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{569 X^{568} Y^{848}}{848 X^{569} Y^{847}} = - \frac{569 Y}{848 X} $$