Let 𝑔 of 𝑥 be equal to 𝑓 of 𝑥
over negative four ℎ of 𝑥 minus five. Given that 𝑓 of negative two is
equal to negative one, 𝑓 prime of negative two is equal to negative eight, ℎ of
negative two is equal to negative two, and ℎ prime of negative two is equal to five,
find 𝑔 prime of negative two.
In this question, we’re asked to
find 𝑔 prime of negative two. So let’s start by differentiating
𝑔 of 𝑥. 𝑔 of 𝑥 is a rational function, so
we’ll need to use the quotient rule. The quotient rule tells us that 𝑢
over 𝑣 prime is equal to 𝑣𝑢 prime minus 𝑢𝑣 prime all over 𝑣 squared. Setting 𝑔 of 𝑥 equal to 𝑢 over
𝑣, we can see that 𝑢 is equal to 𝑓 of 𝑥. And 𝑣 is equal to negative four ℎ
of 𝑥 minus five. 𝑢 prime will be equal to 𝑓 of 𝑥
prime. Now, the prime simply represents a
differentiation with respect to 𝑥. So therefore, 𝑓 of 𝑥 prime is
identical to 𝑓 prime of 𝑥. Next, we need to find 𝑣 prime. So that’s negative four ℎ of 𝑥
minus five prime.
Now, again, since a prime simply
represents a differentiation with respect to 𝑥, we can apply normal differentiation
rules here. And so, differentiating the
constant term negative five will result in zero. So we can say that this is equal to
negative four ℎ of 𝑥 prime. Now, since our function ℎ of 𝑥 is
being multiplied by a constant, negative four. We can use our derivative rules and
take the negative four out of the derivative. Giving us negative four multiplied
by ℎ of 𝑥 prime.
And now, we can apply the same
logic as we did for 𝑓 of 𝑥 prime. And we can say that 𝑣 prime is
equal to negative four ℎ prime of 𝑥. Now, we can substitute into the
quotient rule in order to find 𝑔 prime of 𝑥. Now that we have found 𝑔 prime of
𝑥, we can substitute in 𝑥 is equal to negative two. Now, we have formed an equation in
terms of 𝑓 of negative two, 𝑓 prime of negative two, ℎ of negative two, and ℎ
prime of negative two. All of which we have been given the
value of in the question. And so, we’re able to substitute in
these values here.
Now, our final step in finding 𝑔
prime of negative two is to simplify this. Expanding the brackets, we get
negative 24 minus 20 all over nine. This gives us a solution that 𝑔
prime of negative two is equal to negative 44 over nine.