in this example we're going to have a
look at using the laws of logarithms to
simplify a differentiation which
otherwise we might not be able to
approach. To understand the method used in this video you should already be
familiar with these laws so if you're
not we recommend you take some time to
revise them before watching the rest of
this video. For reference we'll quickly
recap these laws now. The first says that
log of A x B is equal to log A +
log B. The second says that log A/B is equal to log A - log B and
the third that log A to the power m is m
times log A. So let's have a look at this
tricky example, y = x to
the power sinx. There's no obvious way
to differentiate this directly
so, which law for logarithms should we
use to try and help. Well the right-hand
side has the form x to the power of
something, so this looks like the third
law might come in useful if only we had
a logarithm on this right-hand side.
The trick is to take a logarithm of both
sides that way the two sides remain
equal. We take the natural logarithm that
is log to the base. So we now have
lny is equal to lnx to the power
sinx. By law number three the right
hand side can be written as sinx times
lnx. We can now differentiate the right-hand side using the product rule. But what
about the left hand side well, we can
differentiate this side too but we have
to bear in mind that y is a function of
x and here we now have y inside another
function, so we will have to use the
chain rule. So we have d/dx of
lny is equal to d/dx of sinx times lnx.
On the left-hand side we use the
chain rule,
lny differentiates to 1/y but
since y is a function of x we are
differentiating with respect to x we
must times by dy/dx. On the right-hand
side we must use the product rule, so if
we put u is equal to sinx then du/dx
is equal to cosx and if v is
equal to lnx then dv/dx is 1/x.
So using the product rule, the right-hand side becomes sinx times 1/x + lnx
times cosx. Let's tidy  this up,
we can write it as sinx + x times lnx
times cosx all over x. Now we're at
the point where we can isolate what we
really wanted to get at, which is dy/dx. We do this by multiplying both sides
by y. So we have dy/dx = y times sinx + x times lnx times cosx all
over x. But remember we know what y is, it's x to the power sinx, so in fact we have
dy/dx is x to the power sinx times
sinx + x times lnx times cosx
divided by x. In general we don't want to
see any y's on the right-hand side just
x's, which is why we rewrite the y in
terms of x.
