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Find dπ¦/dπ₯, given that π¦ = (4 ln π₯ + 3)/(4 ln π₯ β 7).

Find dπ¦ by dπ₯, given that π¦ 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 dπ¦ by dπ₯. This says that the derivative of the quotient of two differentiable functions π’ and π£ is π£ times dπ’ by dπ₯ minus π’ times dπ£ by dπ₯ all over π£ squared. We let π’ be equal to four times the natural log of π₯ plus three. And π£ is equal to the denominator of our fraction. Thatβs four times the natural log of π₯ minus seven. We then quote the general result for the derivative of the natural log of π₯; itβs one over π₯. And since the derivative of a constant is zero, we see that dπ’ by dπ₯ is equal to four lots of one over π₯, which is simply four over π₯. And similarly dπ£ by dπ₯ is also four over π₯. dπ¦ by dπ₯ is equal to π£ times dπ’ by dπ₯ minus π’ times dπ£ by dπ₯ all over π£ squared.

Letβs multiply the numerator and denominator of this fraction by π₯ to simplify. When we do, we see that dπ¦ 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. Itβs negative 40 over π₯ times four times the natural log of π₯ minus seven squared.

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