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

Find d𝑦/dπ‘₯, given that 𝑦 = (4 ln π‘₯ + 3)/(4 ln π‘₯ βˆ’ 7).

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Video Transcript

Find d𝑦 by dπ‘₯, given that 𝑦 equals four times the natural log of π‘₯ plus three over four times the natural log of π‘₯ 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 dπ‘₯. This says that the derivative of the quotient of two differentiable functions 𝑒 and 𝑣 is 𝑣 times d𝑒 by dπ‘₯ minus 𝑒 times d𝑣 by dπ‘₯ all over 𝑣 squared. We let 𝑒 be equal to four times the natural log of π‘₯ plus three. And 𝑣 is equal to the denominator of our fraction. That’s four times the natural log of π‘₯ minus seven. We then quote the general result for the derivative of the natural log of π‘₯; it’s one over π‘₯. And since the derivative of a constant is zero, we see that d𝑒 by dπ‘₯ is equal to four lots of one over π‘₯, which is simply four over π‘₯. And similarly d𝑣 by dπ‘₯ is also four over π‘₯. d𝑦 by dπ‘₯ is equal to 𝑣 times d𝑒 by dπ‘₯ minus 𝑒 times d𝑣 by dπ‘₯ all over 𝑣 squared.

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

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