We're given the following equation:
In order to find
we must differentiate both sides of the equation.
Lets start differentiating the left side (
):
The
simply servers to let us know we're differentiating whatever follows it (in this case
) with respect to
.
What we used to get the result is called the "power rule for differentiation" it states the following:
In which
is any variable (in the previous case
) and
is any constant (in the previous case this
).
Now we'll differentiate the right side of the equation (
):
What we did to differentiate the right side was, first, apply something called "product rule" for differentiation, it states the following:
In which
and
are arbitrary functions of an arbitrary variable (
) (in this case
and
).
After that we applied something called "chain rule" for differentiation, which states the following:
if
, then
Finally, the
we introduced as a factor after differentiating the right side (we also did it with the left side but with a
) is a consequence of the chain rule, it is always done.
Finally, equaling both differentiated sides of the equation we have:We solve for , and the answer is: