# Question Video: Differentiating Rational Functions Using the Quotient Rule

integrating quotient functions
integrating quotient functions

### Video Transcript

Differentiate the function 𝑓 of 𝑥 which is equal to five 𝑥 squared minus one over seven 𝑥 plus six.

The first thing that stands out to us in this question is that we have the quotient of two differentiable functions, the function five 𝑥 squared minus one in the numerator and the function seven 𝑥 plus six in the denominator. To differentiate functions of this type, we need to use the quotient rule. This tells us that, for two differentiable functions 𝑢 and 𝑣, the derivative of their quotient, 𝑢 over 𝑣, is equal to 𝑣 multiplied by 𝑢 prime minus 𝑢 multiplied by 𝑣 prime all over 𝑣 squared.

Let’s see how to apply the quotient rule in this question. Remember, 𝑢 is the function in the numerator of the quotient and 𝑣 is the function in the denominator. So we’re going to let 𝑢 equal the function in the numerator, five 𝑥 squared minus one, and 𝑣 equal the function in the denominator, seven 𝑥 plus six. We then need to find each of their individual derivatives with respect to 𝑥, which we can do using the power rule of differentiation. 𝑢 prime or d𝑢 by d𝑥 is equal to five multiplied by two 𝑥, which is 10𝑥. And remember, the derivative of a constant, in this case negative one, is just zero. 𝑣 prime or d𝑣 by d𝑥 is equal to seven. So we have both the derivatives of 𝑢 and 𝑣 with respect to 𝑥.

We’re now ready to substitute into the quotient rule to find 𝑓 prime of 𝑥. It’s equal to 𝑣 multiplied by 𝑢 prime. That’s seven 𝑥 plus six multiplied by 10𝑥 minus 𝑢 multiplied by 𝑣 prime. That’s five 𝑥 squared minus one multiplied by seven. And this is all divided by 𝑣 squared. That’s seven 𝑥 plus six all squared. We now just need to distribute the parentheses in the numerator and simplify the result. We obtain 70𝑥 squared plus 60𝑥 minus 35𝑥 squared plus seven all over seven 𝑥 plus six all squared. And then, collecting like terms in the numerator, we have 35𝑥 squared plus 60𝑥 plus seven all over seven 𝑥 plus six all squared.

By applying the quotient rule then, we’ve differentiated our function 𝑓 of 𝑥. Notice that it would also be possible to answer this question using the product rule. If we were to express our function 𝑓 of 𝑥 as five 𝑥 squared minus one multiplied by seven 𝑥 plus six to the power of negative one. And if you’re interested, you could answer the question using the product rule and confirm it does indeed give the same result.

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