This rate of change of the function in the direction L, which we have found as df over dt taken at t also has another name. It is called the directional derivative in the direction L and this is another notation using around d taken at x_0 point. So, once again, this is a sum, where coordinates or the gradient vector multiplied by the corresponding coordinates of the direction L. The only problem is that when we write down parametric equation of the straight line, we use L as the direction vector, but when we change this vector by multiplying it by a factor, for example, we can, instead of L, use double L vector. Nothing changes with the straight line, we have the same straight line. Simply in directional vector has a double length compared with a length L. But this, a multiplication by a vector, changes the directional derivative. So, in order to avoid this ambiguity connected with any length of the directional vector, let's consider,and it will be used further. Only a unit vector, as the directional vector L, so, once again, we'll agree that when we use this formula for directional derivative, the length or the directional vector is always 1. So, this is a univector. If we continue to write down as an inner product, we have a grad, we have L, and we can recall that the inner product of two vectors can be rewritten as the product of the norms or lengths. Don't forget, from now on, the length of L vector is 1 and the products or the lengths or norms is multiplied by the cosine Phi, which is Phi an angle between two vectors. Now, what if we are able to change the direction or the straight line which passes through the x_0 point? Each time there is some angle between the gradient, so, Phi is the angle between the gradient and directional vector L, so if Phi equals 0, the cosine function takes the greatest value and it equals 1. That gives us the result of both the gradient vector. Gradient vector points in the direction or the most rapid growth or the function f. So, this is the fundamental fact about the gradient. How can we use this fact that the gradient vector points direction or the maximum growth or the most rapid growth or the function? Let's consider a production function Q, and for a particular case, so we will consider a Cobb-Douglas function, some particular keys. So, let it be L cubed, which is labor, times capital K squared. There are particular values of labor and capital employed in the technological process. So, let LV1 and K equals 1. Now, where are trying to increase the output of the production by employing extra amounts of labor and capital. So, the question is, what proportion of additional units of labor and capital you have to use in order to increase the output or the production in the most fastest way? Let me draw the coordinate plane, where the horizontal axis is a labor axis and the vertical axis is a capital. Now, we're drawing the isoquant. Remember, isoquant is the curve, the points of which provide the same amount of output. So, whenever you choose a point from the isoquant, you substitute this bundle of vectors into the production function and you get the same amount of output. For instance here, we have a point where labor is 1 and capital is 1 and production equals 1. Now, we need to calculate the gradient vector at this point, the gradient vector at 0.11. Firstly, how to find it, we can do it mentally. We need to differentiate the function with respect to L and substitute the given values of L and K, then we have 3. If we differentiate with respect to capital K and substitute the values, we have 2. So, this is gradient vector, and it looks like that, something like that, although it's more resemble a vector which points at 45 degrees, this vector clearly has some other angle. But anyway, so, we know the gradient shows the direction of the most rapid growth of the production function. So, when we would like to increase the production in the fastest way possible, we need to employ additional units of labor and capital. The arrow which points in the direction of the maximum growth is the hypotenuse of the right angle whose legs are 3 and 2. So, the proportion of the vectors we need to employ additional units of vectors will be Delta L over Delta K equals 3 over 2. So, that's advice for the firm to follow.
