### Video Transcript

Find the first derivative of 𝑦 is
equal to 8𝑥 plus five over three 𝑥 plus 22.

Here, we can see that our function
𝑦 is a rational function. So we can find its derivative by
using the quotient rule. The quotient rule tells us that if
we differentiate the quotient of two functions, so 𝑢 over 𝑣, with respect to
𝑥. Then it’s equal to 𝑣 multiplied by
the differential of 𝑢 with respect to 𝑥 minus 𝑢 timesed by the differential of 𝑣
with respect to 𝑥 all over 𝑣 squared. In order to find the first
derivative of 𝑦, let’s start by labeling 𝑢 and 𝑣 from our equation. 𝑢 will be equal to the numerator
of the function, so eight 𝑥 plus five. And 𝑣 will be equal to the
denominator of the function, so that’s three 𝑥 plus 22.

Next, we must find d by d𝑥 of 𝑢
and d by d𝑥 of 𝑣, or d𝑢 by d𝑥 and d𝑣 by d𝑥. 𝑢 and 𝑣 are both polynomials. So we can simply differentiate them
term by term. Writing 𝑢 in terms of powers of
𝑥, we can say that it’s equal to eight 𝑥 to the power of one plus five 𝑥 to the
power of zero. In order to differentiate, we
simply multiply by the power and decrease the power by one. For the first time, we multiply by
the power, so that’s one, and decrease the power by one, to zero. Leaving us with one timesed by
eight 𝑥 to the power of zero. For the second term, we multiply by
the power, so that’s zero, and decrease the power by one, to negative one. Giving us zero multiplied by five
𝑥 to the negative one.

In the first term, 𝑥 to the power
of zero is just one. So this becomes eight. In the second term, we’re
multiplying by zero. So that term becomes zero. Therefore, we find that d𝑢 by d𝑥
is equal to eight. We can use a similar method to find
d𝑣 by d𝑥. And we find that it’s equal to
three. Now that we found d𝑢 by d𝑥 and
d𝑣 by d𝑥, we’re ready to use the quotient rule. We find that d𝑦 by d𝑥 is equal to
𝑣, which is three 𝑥 plus 22, multiplied by d𝑢 by d𝑥, so that’s eight, minus 𝑢,
so that’s eight 𝑥 plus five, multiplied by d𝑣 by d𝑥, so that’s three. And this is all over 𝑣 squared, so
that’s three 𝑥 plus 22 all squared.

Next, we can expand the
brackets. And then simplify to find that our
solution is that the first derivative of 𝑦 is equal to 161 over three 𝑥 plus 22
all squared.