Write down the utility function and the budget constraint. The utility function, U(x,y), is a function with respect to two goods, x and y. The purpose of utility maximization is to find out how much of each of these to buy.
Write down the budget constraint. This is the amount it will cost to buy x and y, so it depends on the prices Px and Py. A typical budget constraint will look like Px * x + Py * y ≤ I, where I is your income. In other words, the budget constraint is the price of x times the quantity of x added to the price of y times the quantity of y, which cannot be larger than your total income.
Combine the two equations to form the Lagrangian expression, which can be written as L = U(x,y) + λ[I - Px*x - Py*y], where λ is called the Lagrangian multiplier. The calculus steps will all be performed on the Lagrangian.
Take the derivative of the Lagrangian with respect to x and set the resulting equation to 0. This will leave you with dL/dx = MUx - λ * Px = 0. In this case, MUx is the "marginal utility with respect to x," which is the same as the derivative of U(x,y) with respect to x.
Take the derivative of the Lagrangian with respect to y and set the resulting equation to 0. This will leave you with dL/dy = MUy - λ * Py = 0. In this case, MUy is the "marginal utility with respect to y," which is the same as the derivative of U(x,y) with respect to y.
Take the derivative of the Lagrangian with respect to λ and set the resulting equation to 0. This will leave I - Px * x - Py * y = 0, which is just the budget constraint.
Solve for x as a function of y. This can be done by writing MUx = λ * Px and MUy = λ * Py, which can easily be derived from above. Dividing, and canceling out the λs, you are left with MUx/MUy = Px/Py. The left-hand value is the Marginal Rate of Substitution, and the right-hand value is the slope of the indifference curve. Depending on the particular utility function given in the problem, use these values to write x = f(x).
Plug x = f(y) into the budget constraint. The will leave I - Px * f(y) - Py * y = 0. Since this is an equation only in y, solve for y.
Finally, solve for x using the value of y you found. This gives the equation I - Px * x - Py * y. Since Px, Py, and y are all known, solve for x. The values of x and y you have found are the utility maximizing values of the two goods.