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

If 𝑦 = ((𝑥 + 5)/(𝑥 − 5)) − ((𝑥 − 5)/(𝑥 + 5)), find d𝑦/d𝑥.

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### Video Transcript

If 𝑦 is equal to 𝑥 plus five over 𝑥 minus five minus 𝑥 minus five over 𝑥 plus five, find d𝑦 by d𝑥.

Our function, 𝑦, consists of two rational expressions, 𝑥 plus five over 𝑥 minus five and 𝑥 minus five over 𝑥 plus five. And we could find d𝑦 by d𝑥 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 𝑥 plus five squared minus 𝑥 minus five squared all over 𝑥 minus five times 𝑥 plus five. We can expand the brackets and then simplify to obtain that 𝑦 is equal to 20𝑥 over 𝑥 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 𝑢 over 𝑣 prime is equal to 𝑣𝑢 prime minus 𝑢𝑣 prime over 𝑣 squared. Setting 𝑦 equal to 𝑢 over 𝑣, we obtain that 𝑢 is equal to 20𝑥 and 𝑣 is equal to 𝑥 squared minus 25. Next, we can find 𝑢 prime and 𝑣 prime, which gives us that 𝑢 prime is equal to 20 and 𝑣 prime is equal to two 𝑥.

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