Video: Understanding the Quotient Rule

Let 𝑔(π‘₯) = 𝑓(π‘₯)/(βˆ’4β„Ž(π‘₯) βˆ’ 5). Given that 𝑓(βˆ’2) = βˆ’1, 𝑓′(βˆ’2) = βˆ’8, β„Ž(βˆ’2) = βˆ’2, β„Žβ€²(βˆ’2) = 5, find 𝑔′(βˆ’2).

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Video Transcript

Let 𝑔 of π‘₯ be equal to 𝑓 of π‘₯ over negative four β„Ž of π‘₯ minus five. Given that 𝑓 of negative two is equal to negative one, 𝑓 prime of negative two is equal to negative eight, β„Ž of negative two is equal to negative two, and β„Ž 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 π‘₯. 𝑔 of π‘₯ is a rational function, so we’ll need to use the quotient rule. The quotient rule tells us that 𝑒 over 𝑣 prime is equal to 𝑣𝑒 prime minus 𝑒𝑣 prime all over 𝑣 squared. Setting 𝑔 of π‘₯ equal to 𝑒 over 𝑣, we can see that 𝑒 is equal to 𝑓 of π‘₯. And 𝑣 is equal to negative four β„Ž of π‘₯ minus five. 𝑒 prime will be equal to 𝑓 of π‘₯ prime. Now, the prime simply represents a differentiation with respect to π‘₯. So therefore, 𝑓 of π‘₯ prime is identical to 𝑓 prime of π‘₯. Next, we need to find 𝑣 prime. So that’s negative four β„Ž of π‘₯ minus five prime.

Now, again, since a prime simply represents a differentiation with respect to π‘₯, 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 β„Ž of π‘₯ prime. Now, since our function β„Ž of π‘₯ 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 β„Ž of π‘₯ prime.

And now, we can apply the same logic as we did for 𝑓 of π‘₯ prime. And we can say that 𝑣 prime is equal to negative four β„Ž prime of π‘₯. Now, we can substitute into the quotient rule in order to find 𝑔 prime of π‘₯. Now that we have found 𝑔 prime of π‘₯, we can substitute in π‘₯ is equal to negative two. Now, we have formed an equation in terms of 𝑓 of negative two, 𝑓 prime of negative two, β„Ž of negative two, and β„Ž 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.

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