Something more should be said about linear approximations. So, once again, a linear approximation is as follows. So, if Z is a function of x, y, and we would like to approximate the difference between Z and Z0, where we have, we can replace this Delta Z with total differential, so let it be df, where df, as earlier, is the sum, we take the partial derivative with respect to x, multiply by dx, differential of x plus df over dy times dy. So, this Delta Z roughly equals df, and this approximation is the best when dx and dy are quite small. Let's consider some application to production theory. So, given a production function, we would like to evaluate the change in production when labor and capital increases by the increments. So, initially labor was employed in this amount, L, now, we add some additional labor and also we add to capital, the differential of capital. Now, well, using linear approximation to approximate the change in the output. So, Delta Q is replaced by its total differential, and we use the formula, remember the partial derivative of the production function with respect to labor is called marginal product of labor, and the second term starts with MPk which is the marginal product of capital times dk. Now, we'll consider chain rule for multivariate functions, but firstly we need to consider equations of lines in n-dimensional space. Let's consider firstly a two-dimensional space, a Cartesian plane. We start with drawing axis, x and y, and this is a straight line which is completely defined by a fixed point which we'll call original point, and these are the coordinates of this point, x0 and y0, and them directional vector which shows the direction of the straight line, so, we'll denote by L vector, which has two coordinates written in the form of a column vector, let it be Alpha and Beta, using Greek letters here. So, how to define any point which belongs to the straight line. So, given x, y, a point which belongs to the straight line, we can represent this column in the form of the sum of original or initial point plus, we take vector L, which is the direction of the straight line, and multiply by t, where t is so-called parameter, t takes any real values, and this equation is called the parametric equation of the straight line, in R2. This notion of the parametric equation can be easily generalized to the space of n variables because it doesn't matter whether you have only two coordinates or n coordinates. So, in n-dimensional space, the parametric equation straight line, equation of a line, can be written in the same fashion. So, this time a point which belongs to the straight line is a column vector, and in order to define the straight line, we need to indicate a fixed point which is an initial point like in the drawing. In order to indicate it, we'll write zero as a upper-script plus, there should be a directional vector indicating the direction of the straight line, let it be L, as a layer, and this vector is multiplied by a scalar t, which is a parameter. This parameter can take any real values, and this is the general form or the parametric equation of a straight line in n-dimensional space.