Another method by which we can obtain a well-defined, finite number from infinitesimal quantities is to divide one such quantity by another.
We shall assume throughout that we are dealing with well-behaved functions, which means that you can plot the graph of such a function without lifting up your pencil, and you can do the same with each of the function's derivatives. So what is a function, and what is the derivative of a function?
A function
is a machine with an input and an output. Insert a number
and out pops the number
Rather confusingly, we sometimes think of
not as a machine that churns out numbers but as the number churned out when
is inserted.
The (first) derivative
of
is a function that tells us how much
increases as
increases (starting from a given value of
say
) in the limit in which both the increase
in
and the corresponding increase
in
(which of course may be negative) tend toward 0:
![{\displaystyle f'(x_{0})=\lim _{\Delta x\rightarrow 0}{\Delta f \over \Delta x}={df \over dx}(x_{0}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742d2e9b4ff9597ac3c72dfbde8b1aa53685a47e)
The above diagrams illustrate this limit. The ratio
is the slope of the straight line through the black circles (that is, the
of the angle between the positive
axis and the straight line, measured counterclockwise from the positive
axis). As
decreases, the black circle at
slides along the graph of
towards the black circle at
and the slope of the straight line through the circles increases. In the limit
the straight line becomes a tangent on the graph of
touching it at
The slope of the tangent on
at
is what we mean by the slope of
at
So the first derivative
of
is the function that equals the slope of
for every
To differentiate a function
is to obtain its first derivative
By differentiating
we obtain the second derivative
of
by differentiating
we obtain the third derivative
and so on.
It is readily shown that if
is a number and
and
are functions of
then
and ![{\displaystyle {d(f+g) \over dx}={df \over dx}+{dg \over dx}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3d77aff386c587be2fde910d5bfd0d1ebf6942)
A slightly more difficult problem is to differentiate the product
of two functions of
Think of
and
as the vertical and horizontal sides of a rectangle of area
As
increases by
the product
increases by the sum of the areas of the three white rectangles in this diagram:
In other "words",
![{\displaystyle \Delta e=f(\Delta g)+(\Delta f)g+(\Delta f)(\Delta g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba71984c5c64c9b3d820a0f8289b53fcf25d8cc4)
and thus
![{\displaystyle {\frac {\Delta e}{\Delta x}}=f\,{\frac {\Delta g}{\Delta x}}+{\frac {\Delta f}{\Delta x}}\,g+{\frac {\Delta f\,\Delta g}{\Delta x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4841b1d2664c2e3fa7e3b890869e0b196fe0910f)
If we now take the limit in which
and, hence,
and
tend toward 0, the first two terms on the right-hand side tend toward
What about the third term? Because it is the product of an expression (either
or
) that tends toward 0 and an expression (either
or
) that tends toward a finite number, it tends toward 0. The bottom line:
![{\displaystyle e'=(fg)'=fg'+f'g.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7065edec4bb88a33701caa4bae2a7dfad2232b)
This is readily generalized to products of
functions. Here is a special case:
![{\displaystyle (f^{n})'=f^{n-1}\,f'+f^{n-2}\,f'\,f+f^{n-3}\,f'\,f^{2}+\cdots +f'\,f^{n-1}=n\,f^{n-1}f'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e13dc57d30fe071afa2d5b837ab3661b3c7d65c)
Observe that there are
equal terms between the two equal signs. If the function
returns whatever you insert, this boils down to
![{\displaystyle (x^{n})'=n\,x^{n-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfd1e39281ef4e04c64534b4198379c0b264670)
Now suppose that
is a function of
and
is a function of
An increase in
by
causes an increase in
by
and this in turn causes an increase in
by
Thus
In the limit
the
becomes a
:
![{\displaystyle {dg \over dx}={dg \over df}{df \over dx}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fcd5e02a5af80db19c70168a9e02facd88bcfb)
We obtained
for integers
Obviously it also holds for
and
- Show that it also holds for negative integers
Hint: Use the product rule to calculate ![{\displaystyle (x^{n}x^{-n})'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7070de40093d0b30d3ab62b48e8c3faa3487b97)
- Show that
Hint: Use the product rule to calculate ![{\displaystyle ({\sqrt {x}}{\sqrt {x}})'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84e7b2a4a7307d4e722d12ba298101394dabd6e6)
- Show that
also holds for
where
is a natural number.
- Show that this equation also holds if
is a rational number. Use ![{\displaystyle {dg \over dx}={dg \over df}{df \over dx}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fcd5e02a5af80db19c70168a9e02facd88bcfb)
Since every real number is the limit of a sequence of rational numbers, we may now confidently proceed on the assumption that
holds for all real numbers