Since we established in our latest example that
the maximal growth is somehow related to the gradient,
let us try to generalize this concept
for arbitrary function and arbitrarily point.
Firstly, let's start with revisiting
our calculation and growth
for the directional derivative.
Directional derivative equals to
scalar product of the gradient at
a given point and
directional vector in case
the directional vector is normalized.
Also we are going to stress out one more time
that we're particularly interested into
the geometrical sense of
the scalar projects which is the multiplication of
the length of the vector with
the cosine of the angle in between them.
So the question is;
for a fixed point,
fixed function what is the direction of
maximal growth, that's stays forward.
Moreover is if the point is fixed,
then one should expect that partial derivative fixed.
Thus we know the values at
this very point as the gradient is fixed
and then simply known and could not alter them.
The lens of the direction
is pretty much the same which equals to
one thus we couldn't change it either.
So as only thing we are actually able to modify here is
the cosine of the angle in-between
the gradient ends the direction of the differentiation.
So basically the only thing changeable here is this.
So by the definition of the cosine function on
the maximum value of the cosine fraction is one,
and it happens that if the angle between
two vectors equal to zero or vectors are co-directed.
So to sum up what it does mean in terms of
our maximal growth and the direction of maximal growth,
the speed of maximal growth.
The maximum value of
this directional derivative for any point
for every function is a function is differentiable,
is the length of the gradient at this very point,
and the direction of maximal growth
is the gradient itself.
Basically, of course formerly we should
say is that the direction is
the normalized vectors as we need
not only stayed at its core
directed with a gradient but we
should normalize the gradient.
But it doesn't change as there are ideas that at
this very point the gradient shows
not only the direction
where to go to get the maximal value,
but how fast your function changes in around there.
So let us consider for example some basic function
like that equal to X-squared plus Y-squared.
So that's our parabola which has
resulted after the rotation of axons with
state at which the first meeting
of this factor in the earliest weeks.
So first let us start with
considering as a gradient of this function which is 2_X,
2_Y thus for any possible point A, B.
So direction of maximum growth at this point,
I'm going to write L max is well basically 2A,
2B divided by the length of this vector.
It's complicated, and as the value of the derivative on
this direction is two square root
of A squared plus B square.
That was easier.
By symmetric effects,
the direction of maximal growth is also
state as a direction of
minimal growth or maximal decrease.
If you will it is minus,
gradient or inter gradient,
and we're going to use it in the following slide.