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

If 𝑦 = ((π‘₯ + 5)/(π‘₯ βˆ’ 5)) βˆ’ ((π‘₯ βˆ’ 5)/(π‘₯ + 5)), find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 is equal to π‘₯ plus five over π‘₯ minus five minus π‘₯ minus five over π‘₯ plus five, find d𝑦 by dπ‘₯.

Our function, 𝑦, consists of two rational expressions, π‘₯ plus five over π‘₯ minus five and π‘₯ minus five over π‘₯ plus five. And we could find d𝑦 by dπ‘₯ by using the quotient rule on these two rational expressions. However, this would require using the quotient rule twice. We can make our work a little easier by combining the two rational expressions into one. We obtain that 𝑦 is equal to π‘₯ plus five squared minus π‘₯ minus five squared all over π‘₯ minus five times π‘₯ plus five. We can expand the brackets and then simplify to obtain that 𝑦 is equal to 20π‘₯ over π‘₯ squared minus 25.

And now, our function consists of only one rational expression. We’re ready to use the quotient rule to differentiate this function. The quotient rule tells us that 𝑒 over 𝑣 prime is equal to 𝑣𝑒 prime minus 𝑒𝑣 prime over 𝑣 squared. Setting 𝑦 equal to 𝑒 over 𝑣, we obtain that 𝑒 is equal to 20π‘₯ and 𝑣 is equal to π‘₯ squared minus 25. Next, we can find 𝑒 prime and 𝑣 prime, which gives us that 𝑒 prime is equal to 20 and 𝑣 prime is equal to two π‘₯.

Now, we can substitute them into the quotient rule in order to find that d𝑦 by dπ‘₯ is equal to π‘₯ squared minus 25 times 20 minus 20π‘₯ times two π‘₯ all over π‘₯ squared minus 25 squared. We simplify this to obtain that d𝑦 by dπ‘₯ is equal to negative 20π‘₯ minus 500 all over π‘₯ squared minus 25 squared.

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