Video: Understanding the Quotient Rule

Let π(π₯) = π(π₯)/(β4β(π₯) β 5). Given that π(β2) = β1, πβ²(β2) = β8, β(β2) = β2, ββ²(β2) = 5, find πβ²(β2).

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Video Transcript

Let π of π₯ be equal to π of π₯ over negative four β of π₯ minus five. Given that π of negative two is equal to negative one, π prime of negative two is equal to negative eight, β of negative two is equal to negative two, and β prime of negative two is equal to five, find π prime of negative two.

In this question, weβre asked to find π prime of negative two. So letβs start by differentiating π of π₯. π of π₯ is a rational function, so weβll need to use the quotient rule. The quotient rule tells us that π’ over π£ prime is equal to π£π’ prime minus π’π£ prime all over π£ squared. Setting π of π₯ equal to π’ over π£, we can see that π’ is equal to π of π₯. And π£ is equal to negative four β of π₯ minus five. π’ prime will be equal to π of π₯ prime. Now, the prime simply represents a differentiation with respect to π₯. So therefore, π of π₯ prime is identical to π prime of π₯. Next, we need to find π£ prime. So thatβs negative four β 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 of π₯ prime. Now, since our function β 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 of π₯ prime.

And now, we can apply the same logic as we did for π of π₯ prime. And we can say that π£ prime is equal to negative four β prime of π₯. Now, we can substitute into the quotient rule in order to find π prime of π₯. Now that we have found π prime of π₯, we can substitute in π₯ is equal to negative two. Now, we have formed an equation in terms of π of negative two, π prime of negative two, β of negative two, and β prime of negative two. All of which we have been given the value of in the question. And so, weβre able to substitute in these values here.

Now, our final step in finding π 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 π prime of negative two is equal to negative 44 over nine.