Video Transcript
Determine d𝑦 by d𝑥, given that 𝑦
is equal to eight times the sin of four 𝑥 all to the power two 𝑥.
We’re asked to find the derivative
of 𝑦 with respect to 𝑥. But if we look at our function 𝑦,
we have an exponent two 𝑥, which is an expression in 𝑥. And this means that we can’t apply
the usual rules of differentiation, the product, quotient, or chain rules,
directly. What we can do, however, is use
logarithmic differentiation. This is a four-step process for a
function 𝑦 is 𝑓 of 𝑥, where step one is to apply the natural logarithm to both
sides, recalling that the natural logarithm is the log to the base 𝑒 where 𝑒 is
Euler’s number, which is approximately 2.71828 and so on.
In our case, we have 𝑦 is equal to
eight times the sin of four 𝑥 all to the power two 𝑥, and so we take the natural
logarithm on both sides. At this point, we are to specify
that 𝑦 has to be greater than zero. That’s because the log of zero is
undefined and the log doesn’t exist for negative values. If we do want to include negative
values, we must include also the absolute value signs around 𝑦 and 𝑓 of 𝑥 and, in
this case, specify that 𝑦 is nonzero. For our problem, however, we’ll
simply specify that 𝑦 is greater than zero.
Now, our second step in logarithmic
differentiation is where we start to see how taking logs can help us. We use the laws of logarithms to
expand or simplify our right-hand side. And since in our argument for the
logarithm on the right-hand side we have an exponent, we’re going to use the power
rule for logarithms. And this says that log to the base
𝑎 of 𝑏 raised to the power 𝑐 is 𝑐 times log to the base 𝑎 of 𝑏. That is, we bring the exponent 𝑐
down in front of the log and multiply by it. In our case, this translates to the
natural logarithm of 𝑦 is two 𝑥 times the natural logarithm of eight sin four
𝑥. And now we see that on our
right-hand side, we have the product of two expressions in 𝑥.
And this brings us to our third
step in logarithmic differentiation, that’s differentiate both sides with respect to
𝑥. And since we have a product on our
right-hand side, we can use the product rule for differentiation. This says that if 𝑢 and 𝑣 are
differentiable functions of 𝑥, d by d𝑥 of 𝑢𝑣, the product, is 𝑢 times d𝑣 by
d𝑥 plus 𝑣 times d𝑢 by d𝑥. So if we now let 𝑢 equal two 𝑥
and 𝑣 be the natural algorithm of eight times the sin of four 𝑥, then to use the
product rule, we need to find d𝑣 by d𝑥 and d𝑢 by d𝑥. d𝑢 by d𝑥 is two by
sight. And to differentiate 𝑣, we can use
the result that d by d𝑥 of the natural logarithm of a differentiable function 𝑓 of
𝑥 is one over 𝑓 of 𝑥 times d𝑓 by d𝑥.
So d𝑣 by d𝑥 is one over eight sin
four 𝑥 times d by d𝑥 of eight sin four 𝑥. To differentiate eight sin four 𝑥,
we use the fact that d by d𝑥 of sin 𝑢, where 𝑢 is a differentiable function of
𝑥, is d𝑢 by d𝑥 times cos 𝑢 so that d𝑣 by d𝑥 is one over eight times the sin of
four 𝑥 all times four times eight cos four 𝑥. We have a common factor of eight on
the denominator and the numerator which cancel each other out so that d𝑣 by d𝑥 is
four cos four 𝑥 over sin four 𝑥. So we have 𝑢 is two 𝑥 and d𝑢 by
d𝑥 is two, 𝑣 is the natural algorithm of eight sin four 𝑥, and d𝑣 by d𝑥 is four
cos four 𝑥 over sin four 𝑥.
And to simplify this even further
since cos over sin is the cot, we can say d𝑣 by d𝑥 is four times the cot of four
𝑥. So now applying the product rule
for differentiation to our right-hand side, we have two 𝑥, which is 𝑢, times four
times the cot of four 𝑥, which is d𝑣 by d𝑥, plus the natural logarithm of eight
sin four 𝑥, which is 𝑣, times two, which is d𝑢 by d𝑥. And now making some room, we can
rewrite this and rearrange it. We have d by d𝑥 of the natural
logarithm of 𝑦 is equal to two times the natural logarithm of eight sin four 𝑥
plus eight 𝑥 times the cot of four 𝑥.
We’re not quite finished with our
third step since we still need to differentiate the natural logarithm of 𝑦. And again, we can use our known
result that d by d𝑥 of the natural logarithm of a function 𝑓 of 𝑥 is one over 𝑓
of 𝑥 times d𝑓 by d𝑥. And since our 𝑦 is indeed a
function of 𝑥, our left-hand side is simply one over 𝑦 d𝑦 by d𝑥. And this brings us to our final
step in logarithmic differentiation which is to solve for d𝑦 by d𝑥.
We can do this by multiplying both
sides by 𝑦 so that on the left-hand side, the 𝑦’s cancel each other out, and we’re
left with d𝑦 by d𝑥. On our right-hand side, we can
reintroduce our function 𝑦 so that we’re multiplying by eight sin four 𝑥 to the
power two 𝑥 and taking out a common factor of two. Using logarithmic differentiation
to differentiate the function 𝑦 is eight sin four 𝑥 all to the power two 𝑥, we
have d𝑦 by d𝑥 is two times eight sin four 𝑥 to the power two 𝑥 multiplied by the
natural logarithm of eight sin four 𝑥 plus four 𝑥 times the cot of four 𝑥.
