I want to make some remark concerning notations. We always use df over dx, df over dy to represent partial differentiation with respect to x or y, but when we write these derivatives very often and if we have some lengthy formulas it's kind of boring to write all the time these quotients with round d's, and there is a shorthand notation for both derivatives. So, df over dx can be written as f, and here I will write x representing that this is derivative with respect to x. In some textbooks, there is always a prime symbol here above f, but there is no need actually. So, this is a very good notation because it's compact. The same true for df over dy. df over dy can be written as fy. Now, back to the chain rule. Let's consider the function Z, f of x, y, and x is a function of uv variables. That will be particular function u times v, and as for y, y will be u over v. Now, the question is how to find derivatives, df over du, and later on df over dv. We can use also this shorthand, which will facilitate our writing. So, remember the formula that was in the box. Okay let me write it here. So, df over du is df over dx, times dx over du, plus df over dy, times dy over du. We need to multiply by the derivative dx over du and that will be v plus fy. Our y should differentiated with respect to u, and this is one over v, and the next is df over dv. So, firstly we differentiate with respect to x, the given function. Now, we'll take x, this is our function x, we differentiate it with respect to v, we get u plus fy, and we multiply by the derivative with respect to v, these are the denominator so that's how we get minus u over v squared, and this is the end of our calculations. Now, it's time to work out some basic theoretical facts we haven't discussed earlier. Remember I said that given a function of one variable, we call such a function f of x as a continuously differentiable function if its first derivative is continuous and we indicate this property of the function by the symbol belongs to, belongs to class of continuously differentiable functions. Now, given a function of many variables, so let y be f of x1, x2,...xn. We say that this function belongs to the class of continuously differentiable functions which is C1, if all first order partial derivatives are continuous functions. If we don't indicate within the brackets that was the beginning, a single variable function, sometimes the domain is not the set of all real numbers but probably an open interval. So, we indicated within the brackets. The same true for the n-dimensional keys. So, if we write here, as an example, D, which is an open set in Rn space, then we know that what it means. It means that all partial derivatives are continuous in D. In applications for economics, quite often we do the functions with this property that the functions belong to C1 class of functions. Why it's important, it's important because when we were talking about linear approximations, actually and not necessarily a plane which touches the surface or the function can be a good approximation for the values of the function in question. For example, let's consider a function Z, the absolute value of the sum x plus y. So, let's consider the graph of this function or rather we can talk about the possibility of finding the tangent plane to the graph of this function at the origin. So, interested in tangent plane. The problem is that how we find the derivatives of the partial derivatives of this function at the origin. They simply do not exist as we understand. For example, if we are looking for dz over dx derivative, we need to take the zero value for y and what happens then, this is the absolute value of x and the derivative at zero doesn't exist at the point zero, zero. The same true for another derivative, dz over dy. The same true also doesn't exist. So, that tells us that the tangent plane to the graph of this function at the origin doesn't exist, but sometimes we can calculate the values of the partial derivatives and we can substitute into the equation of the tangent plane. Remember, we need to find dz over dx, dz over dy and substitute into the formula. Still, if we find the equation of the tangent plane, not necessarily the values of z will nicely approximate the values of the function we are investigating, not the properties of. I'll provide an example.
