What are Derivatives? Video Lecture | Mathematics for Grade 11

so what is a derivative anyway
derivatives are functions that describe
how quickly other functions change if a
function around some value of X moves up
in value quickly as the values of X grow
then it could be said that the
derivative of this function is large and
positive around those values of X
if a function stays about the same as X
grows around some neighborhood of X
values then it could be said that the
derivative of this function stays close
to zero around those values of X if for
some neighborhood of X values the
function decreases in value as X grows
then the derivative of such a function
is negative in that neighborhood of X
values you might be concerned that in a
neighborhood of X the function doesn't
just go up or down but it has hills and
valleys of all sorts to keep things
simple we will only consider functions
such that if we choose a narrow enough
neighborhood of X then the values of the
function only go up or down the
neighborhood that we will consider will
be so very tiny that within the function
itself it will lose almost all of its
curviness
and become almost completely a straight
line if we take a look at the
neighborhood of the function f of X at
some value X the size of the
neighborhood will be Delta X in width it
may notice that as long as we keep our
distances very small it is true that if
you increase the size of Delta X the
size of Delta f of X also increases
proportionally for this reason we can't
simply talk about the size of Delta f of
X but we must consider the proportion of
the size of Delta f of X to the size of
Delta X this is another way of saying
that we are interested in the ratio
Delta f of X over Delta X which is also
called the rate of change
we could consider all kinds of other
relationships between these two values
such as delta x over delta f of X or arc
tan Delta f of X over Delta X or some
other relationship but the most useful
measure of the slope in the later
applications of calculus happens to be
Delta f of X over Delta X if we make the
two numbers infinitesimally small then
all that we are left with is the exact
rate of change at that particular
position which is going to be called the
derivative and the function that
provides the value of the derivative at
the position of X is written F prime of
X how do we know that the ratio doesn't
just disappear and become zero or
infinity as the Delta X and Delta f of X
are shrunk this has to do with something
called limits that are more formal
calculus book covers in greater detail
but for us it is sufficient to imagine
that the two sides of a triangle
describing the rate of change have the
same proportions to each other
regardless of how small the triangle
itself becomes therefore if f of X has a
derivative then F prime of X is equal to
Delta f of X over Delta X where Delta f
of X and Delta X are infinitesimally
small at each particular value of X