Back to the straight line, this parametric equation can be written in a vector form like this. Now, what happens if we take the function either N dimensional space and substitute X, which is a point belonging to the straight line inside the function. So, we replace X with the X which is a point in the straight line. So, then Y equals F and here we have zero plus LT. Now, this is a fixed point. L is a direction on the line and T is the venerable. Now, we're introducing the rate of change of the function in direction L. The definition is as follows, this rate of change is given by the derivative. So, we take derivative of this composite function because we have inserted values. This is a vector which belongs to the straight line. So, we're considering an ordinary derivative with respect to T. Now, at what point this derivative should be taken at t equals zero or rather at the point X zero. Now, we will try to find out how to calculate this ordinary derivative. In order to do that, we'll finish the statement here. But firstly, let me derive the expression we're looking for. We'll base our calculation on total differential. So, total differentials we know is df represented. This time we're using the symbol of summation, because we're dealing with a N dimensional space. Here each partial derivative with respect to corresponding coordinate XI, is multiplied by the differential of the independent variable. So, if we take the total differential N, divide the left hand side and the right hand side by the increment in parameter T, what do we get then? Left hand side, DF over DT, this is exactly the rate we're looking for. Here will have. But don't forget we have the parametric equation of the straight line. So, when we take the Ith coordinate or this equation and differentiate it with respect to T, quite often T is regarded as time. Then, what do we get from this formula? What we get is the Ith component or the directional vector L. We may continue here, and we'll get the sum. Now, let us recall the formula which gives them in the product of two vectors. Given RN space when we multiply as an inner product of two vectors X and Y, we use the formula. So, these are two vectors. We calculate the inner product by choosing Ith coordinates of both vectors and after multiplying them, we sum up from I one up to N. So, that's the formula for inner product. When we compare it with this formula, we see that this is also an inner product. Now, what are the vectors? As for L vector, we dealt with respect to earlier and this is a directional line which passes through X zero point. As for the components of this vector, these are components of the vector which is widely used in multidimensional calculus. It is also used in a lot of applications. It is called gradient of the function F. So, let's provide a definition. So, this is important concept gradient where X is a vector. So, by definition, gradient is a vector and we can denote it with the word grad. This is a vector whose coordinates are partial derivatives and they are calculated at some point. Actually here, when we're discussing the rate of change of the function, we are considering X zero point. So, back to the definition this is a vector written in the form of a row. Middle partial derivatives taken at X zero point, this is the first derivative with respect to X one. We continue until we reach the Nth, and this is exactly the vector which is in the formula. So, we can write here a triangular bracket grad fl. So, going back to this formula, we see that we've got our answer. I'm simply rewriting from there, but I have more space here to indicate that this is a gradient calculated at X zero point. So, here we have grad F.