Video: Differentiating a Combination of Rational Functions at a Point Using the Quotient Rule

Evaluate 𝑓′(3), where 𝑓(π‘₯) = (π‘₯/(π‘₯ + 2)) βˆ’ ((π‘₯ βˆ’ 3)/(π‘₯ βˆ’ 2)).

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Video Transcript

Evaluate 𝑓 prime of three, where 𝑓 of π‘₯ is equal to π‘₯ over π‘₯ plus two minus π‘₯ minus three over π‘₯ minus two.

Now, our function is the difference of two rational expressions. We can start by combining the two rational expressions into one. We obtain that 𝑓 of π‘₯ is equal to negative π‘₯ plus six all over π‘₯ squared minus four. And we have written 𝑓 as a rational function. And we’re ready to use the quotient rule, which tells us that 𝑒 over 𝑣 prime is equal to 𝑣𝑒 prime minus 𝑒𝑣 prime all over 𝑣 squared. Setting 𝑓 of π‘₯ equal to 𝑒 over 𝑣, we obtain that 𝑒 is equal to negative π‘₯ plus six, and 𝑣 is equal to π‘₯ squared minus four. We then find that 𝑒 prime is equal to negative one, and 𝑣 prime is equal to two π‘₯.

Substituting 𝑒, 𝑣, 𝑒 prime, and 𝑣 prime back into the quotient rule. We find that 𝑓 prime of π‘₯ is equal to π‘₯ squared minus four multiplied by negative one minus negative π‘₯ plus six multiplied by two π‘₯ all over π‘₯ squared minus four squared. In order to find 𝑓 prime of three, we simply substitute π‘₯ equals three into 𝑓 prime of π‘₯. We obtain that 𝑓 dash of three is equal to three squared minus four multiplied by negative one minus negative three plus six multiplied by two times three all over three squared minus four squared. Which simplifies to negative five minus 18 over 25. This gives us a solution that 𝑓 prime of three is equal to negative 23 over 25.

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