We'll be discussing differentiation of functions of many variables. We start with a refreshing of the ordinary derivatives. So, let's consider a function of one variable. So, x is an independent variable, y is dependent variable, and x belongs to some interval. Let's suppose we have a point which belongs to the same interval, and we consider a limit, this is a quotient where in the numerator we have, and in the denominator delta x, where delta x is an increment in x. So, we consider this limit as delta x tends to zero, and if this limit exists, we call it derivative of the function. We write f prime, and derivative is taken at x0 point. Now, we deal with a function which depends on two independent variables x and y, and dependent variable will be denoted by Z. So, z is a function of x and y. So, we're introducing so called partial differentiation. Partial differentiation. We're choosing Z because we usually label axes in three dimensional space. So, we have x axis, y axis, and z axis. So, let's suppose this function is defined in some open set. So x, y, a point in two dimensional plane belongs to D, where D is an open set in R2, our cartesian plane. So, this function is defined and also we take another point, a fixed point, which belongs to the same search D. This is not a rigorous definition at the moment, we're just approaching it. So, what if we keep y value fixed at the level of y0 but as for x value we'll slightly change it from X0 to another value which is x. That means that as before in a single variable case we'll have increment of x which is delta x. So, x is x0 plus delta x then. As I said, y will be kept at the value of y0. Now, let's consider a very similar limit, a limit of the quotient. But this time since we deal with a function depending on two variables x and y, this quotient will take a slightly different look. So, we're considering the limit as delta x approaches or the quotient, a long line. Here, we take f at the point x0 plus delta x, y0 is the value of the second variable, we subtract the value of the function taken at the original point x0,y0, and as earlier, we divide by delta x. Now, if this limit exists then we say that this function is differentiable with respect to x at a point x0, y0, and what's more we use a special notation for this partial derivative. So, we have found partial derivative of the function this time with respect to x. So, the notation will be as follows, I'm using instead of straight D, straight D was used to denote ordinary derivative. So, this is a round D, df over dx, and here we write the point at which this value was found or taken. What about differentiation with respect to another variable Y? We'll do it in a similar fashion. So, let's suppose we are considering a limit, this time increment of y tends to zero, as for x value it will be fixed at the level of a zero. Now, we are considering a quotient where in the denominator we have delta y, and here we have f, x0, y0 plus delta y, and we're subtracting that value at original point x0, y0. Now, if this limit exists, we call it a partial derivative of the function f with respect to y and we use notation, this is df over dy evaluated at the point x0, y0.