# Video: Finding the Unknowns in a Rational Function given Its Value and the Value of Its First Derivative at a Point

Suppose that π(π₯) = (π₯Β² + ππ₯ + π)/(π₯Β² β 7π₯ + 4). Given that π(0) = 1 and πβ²(0) = 4, find π and π.

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### Video Transcript

Suppose that π of π₯ is equal to π₯ squared plus ππ₯ plus π all over π₯ squared minus seven π₯ plus four. Given that π of zero is equal to one and π prime of zero is equal to four, find π and π.

Our first step in this question can be to substitute π₯ equals zero into π of π₯. Since weβre given that π of zero is equal to one. We obtain that π of zero is equal to zero squared plus π times zero plus π all over zero squared minus seven times zero plus four. Now, all of these terms will go to zero apart from π and four. Weβre left with π of zero is equal to π over four. Next, we use the fact that the question has told us that π of zero is equal to one. And so, we can set this equal to one. From this, we find that π is equal to four. Next, we can use the fact that π prime of zero is equal to four. However, first of all, we must find π prime of π₯. In order to do this, we need to differentiate π. Since π is a rational function, we can use the quotient rule in order to find its derivative.

The quotient rule tells us that π’ over π£ prime is equal to π£ times π’ prime minus π’ times π£ prime all over π£ squared. Setting our function π of π₯ equal to π’ over π£, we obtain that π’ is equal to π₯ squared plus ππ₯ plus π. And π£ is equal to π₯ squared minus seven π₯ plus four. We can find π’ prime and π£ prime by differentiating these two functions. Giving us that π’ prime is equal to two π₯ plus π and π£ prime is equal to two π₯ minus seven. Now, we can substitute these into the quotient rule. We obtain that π prime of π₯ is equal to π₯ squared minus seven π₯ plus four multiplied by two π₯ plus π minus π₯ squared plus ππ₯ plus π multiplied by two π₯ minus seven all over π₯ squared minus seven π₯ plus four all squared.

Now, we could simplify π prime of π₯ at this point. However, weβre going to be substituting in π₯ is equal to zero. And so, a lot of these terms would just disappear. Letβs simply substitute π₯ equals zero in here. We obtain this. However, a lot of the terms will vanish to zero, which leaves us with four π plus seven π all over 16. Now, we have found that π is equal to four earlier. And so, we can substitute this in, giving us four π plus 28 all over 16.

Since the question has told us that π prime of zero is equal to four, we can set this equal to four. Then, we simply rearrange this in order to solve for π. Now, we obtain our solution that π is equal to nine. Weβve now found the values of both π and π, which completes the solution to this question.