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

Find d𝑦/d𝑥, given that 𝑦 = (4 ln 𝑥 + 3)/(4 ln 𝑥 − 7).

01:43

### Video Transcript

Find d𝑦 by d𝑥, given that 𝑦 equals four times the natural log of 𝑥 plus three over four times the natural log of 𝑥 minus seven.

In this question, we have a fraction or a quotient. This tells us we can use the quotient rule to find the derivative d𝑦 by d𝑥. This says that the derivative of the quotient of two differentiable functions 𝑢 and 𝑣 is 𝑣 times d𝑢 by d𝑥 minus 𝑢 times d𝑣 by d𝑥 all over 𝑣 squared. We let 𝑢 be equal to four times the natural log of 𝑥 plus three. And 𝑣 is equal to the denominator of our fraction. That’s four times the natural log of 𝑥 minus seven. We then quote the general result for the derivative of the natural log of 𝑥; it’s one over 𝑥. And since the derivative of a constant is zero, we see that d𝑢 by d𝑥 is equal to four lots of one over 𝑥, which is simply four over 𝑥. And similarly d𝑣 by d𝑥 is also four over 𝑥. d𝑦 by d𝑥 is equal to 𝑣 times d𝑢 by d𝑥 minus 𝑢 times d𝑣 by d𝑥 all over 𝑣 squared.

Let’s multiply the numerator and denominator of this fraction by 𝑥 to simplify. When we do, we see that d𝑦 by d𝑥 is equal to four times four times the natural log of 𝑥 minus seven minus four times four times the natural log of 𝑥 plus three all over 𝑥 times four times the natural log of 𝑥 minus seven squared. We distribute the parentheses on our numerator. And we see that we have 16 times the natural log of 𝑥 minus 16 times the natural log of 𝑥 which gives us zero. And we found the derivative of our quotient. It’s negative 40 over 𝑥 times four times the natural log of 𝑥 minus seven squared.