Video Transcript
Find 𝑑𝑦 𝑑𝑥, given that 𝑦
equals the natural logarithm of four 𝑥 plus five to the power of seven.
So the first thing that we’re
looking to do when we differentiate our function is actually rewrite it in another
way. And we can do that because of one
of the log laws. And that law states that if we’ve
got log to the base 𝑏 of 𝑥 to the power of 𝑛, then this can be rewritten as 𝑛
log to base 𝑏 of 𝑥. So when we rewrite our function, we
actually get 𝑦 is equal to seven ln four 𝑥 plus five. So we did that using the log
law.
So now if we’re actually looking to
find 𝑑𝑦 𝑑𝑥, we can say it’s gonna be equal to seven multiplied by 𝑑𝑑𝑥 of ln
four 𝑥 plus five. So what we wanna do now is actually
differentiate our ln four 𝑥 plus five. And the way to do that is actually
by using the chain rule. And the chain rule actually states
that 𝑑𝑦 𝑑𝑥 is equal to 𝑑𝑦 𝑑𝑢 multiplied by 𝑑𝑢 𝑑𝑥. So we can actually adapt this and
use this to solve our problem.
So if we’re looking to
differentiate ln four 𝑥 plus five, we can actually say that 𝑢 is gonna be equal to
four 𝑥 plus five, which is the actual expression inside the parentheses, and 𝑦 is
gonna be equal to ln 𝑢. So first of all, what we’re
actually gonna do is differentiate 𝑢. So it’s 𝑑𝑢 𝑑𝑥, which is just
gonna give us four, because if we differentiate four 𝑥, we get four. And if we differentiate positive
five, it actually just turns to zero. So great! 𝑑𝑢 𝑑𝑥 is equal to four.
So now we’re gonna find 𝑑𝑦
𝑑𝑢. And to do that, we’re actually
gonna differentiate ln 𝑢. We’re going to get one over 𝑢. And we get that because actually
there’s a general form that tells us that if 𝑦 is equal to ln 𝑥, then 𝑑𝑦 𝑑𝑥 is
equal to one over 𝑥.
Okay, great! So now we’ve actually
differentiated both parts. We can use the chain rule to put it
together. So we’re gonna get that the
differential of ln four 𝑥 plus five is equal to four, because that was 𝑑𝑢 𝑑𝑥
multiplied by one over 𝑢, which is 𝑑𝑦 𝑑𝑢. So now just substitute in our value
for 𝑢, which gives us four over four 𝑥 plus five.
Okay, great! So now we’ve actually
differentiated our ln four 𝑥 plus five. There’s just one more step to do to
actually find our final 𝑑𝑦 𝑑𝑥. So we’re gonna get 𝑑𝑦 𝑑𝑥 is
equal to seven multiplied by four over four 𝑥 plus five. And that’s, as we said before, was
because we actually differentiated ln four 𝑥 plus five. So therefore we can say that, given
that 𝑦 equals ln four 𝑥 plus five to the power of seven, 𝑑𝑦 𝑑𝑥 is equal to 28
over four 𝑥 plus five.
It’s worth mentioning at this point
that we could’ve actually used the formula to help us differentiate ln four 𝑥 plus
five. And that would have been that if it
was in the form 𝑦 equals ln 𝑓 𝑥, then 𝑑𝑦 𝑑𝑥 is equal to the differential of
𝑓 𝑥 over 𝑓 𝑥. However, I just wanted to
demonstrate how the chain rule is used to actually get to this result. And we can actually double-check it
using this, because if we actually look, our function, so our 𝑓 𝑥, was actually
four 𝑥 plus five.
So, therefore, if we differentiate
this, we just get four. So that would have been our
numerator. And then on the denominator, it
just would’ve been the function itself, which was four 𝑥 plus five. And as we can see in the previous
step, just before the final answer, this was in fact the value that we found when we
differentiated ln four 𝑥 plus five using the chain rule.
