Let's consider some examples. I'll choose a function f of (x,y) it will be a polynomial function 3x squared y plus 5xy cubed. Let's find both partial derivatives at a given point. The point will be we're taking x_0 one and y_0 also one, okay. So, how to find firstly df over dx taken at (1,1) point. According to our definition we need to substitute for y one. Then this function becomes a function of x alone, and after that we differentiate it as we do with a single variable function. We apply all rules we are familiar with like, derivative of a sum of the difference of the product and a quotient. Here we have after substituting one for y we have a function consisting of two terms. This term is x squared y is one as I said, and this is a number three. So, we apply the rule of differentiation when we differentiate the sum, sum rule, and also we have a factor three, and here we have vector five. So that means that as we know when we differentiate a function which is multiplied by scalar, this scalar can be taken out. All in all we have, I can write like this, 3x squared plus 5x, and here is the prime symbol indicating the derivative that equals 6x plus five. As for the partial derivative with respect to y, we do a similar thing. This time we substitute one for x and differentiate. But, what's more interesting we can find derivative at any point (x,y). Let's do it and let's find df over dy at any point. Remember, we differentiate with respect to y and we keep x fixed. So x is considered as a constant. Now, we get df over dy at any point (x,y) equals. So, we're differentiating with respect to y this is a constant. So, when we differentiate the first term what's left, 3x squared plus second term three goes out because this is a power function, and we have 15xy squared. Another example, this time we'll apply the chain rule. Let us consider a function g of (x,y) which is the square root of x squared plus y squared. Now the question is how to find dg over dx at any point (x,y). So, sometimes we omit using (x,y) within the brackets. This symbol will clearly indicate that the partial derivative is taken at any point with coordinates x and y by default. So, chain rule, y is considered a constant. So, according to the chain rule firstly we differentiate the outer function, this is the square root. Then we clearly get a quotient where in the denominator we have two, and the square root of the same combination x squared plus y squared. Now, in the numerator we differentiate what's inside under the sign of the square root. Here we have 2x. Well, this number two can be crossed out here and there, and the final expression is. Now, I will continue with some examples from economics. All we know in microeconomics the production function, provides the quantity produced as a function of the arguments of this function will be considering two factors of production, labor and capital. According to microeconomics, the derivative, partial derivative with respect to labor is called the Marginal Product of Labor. So, abbreviation is MPL and this definition. If we differentiate the production function with respect to the K, we get another marginal products. This time marginal product of capital. Both derivatives are widely used in the production theory. Let's consider a particular production function which is called Cobb-Douglas, Cobb-Douglas function. In particular we have f, l raised to the power Alpha times k to the power of Beta where these powers are positive numbers. Then MPL, is the derivative is taken at any point in the space of the factors of production, labor and capital. So, we have Alpha l and also MPK.