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Question Video: Finding the First Derivative of a Function Defined by Parametric Equations Involving Logarithmic Functions Mathematics • Higher Education

Given that ๐‘ฅ = 5๐‘ก โˆ’ 4 ln ๐‘ก and ๐‘ฆ = 4๐‘ก + 5 ln 3๐‘ก, find d๐‘ฆ/d๐‘ฅ.

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Video Transcript

Given that ๐‘ฅ equals five ๐‘ก minus four times the natural log of ๐‘ก and ๐‘ฆ equals four ๐‘ก plus five times the natural log of three ๐‘ก, find d๐‘ฆ by d๐‘ฅ.

Here weโ€™ve been given a pair of parametric equations. These are equations for ๐‘ฅ and ๐‘ฆ in terms of a third parameter. Thatโ€™s ๐‘ก. And weโ€™re looking to find the derivative of ๐‘ฆ with respect to ๐‘ฅ. And so, we recall that as long as d๐‘ฅ by d๐‘ก is not equal to zero, the derivative of ๐‘ฆ with respect to ๐‘ฅ is equal to d๐‘ฆ by d๐‘ก divided by d๐‘ฅ by d๐‘ก. And so, it should be quite clear to us that weโ€™re going to need to differentiate each of our respective equations with respect to ๐‘ก. And weโ€™ll do this term by term.

The first term in our equation for ๐‘ฅ is five ๐‘ก. Now, the derivative of five ๐‘ก with respect to ๐‘ก is five. Next, we recall that the derivative of the natural log of ๐‘Ž๐‘ฅ, where ๐‘Ž is some constant, with respect to ๐‘ฅ is one over ๐‘ฅ. So, the derivative of the natural log of ๐‘ก is one over ๐‘ก. And so, the derivative of negative four times the natural log of ๐‘ก is negative four over ๐‘ก.

Letโ€™s repeat this process for d๐‘ฆ by d๐‘ก. The derivative of four ๐‘ก with respect to ๐‘ก is four. Then, the derivative of the natural log of three ๐‘ก must be one over ๐‘ก. So, five times the natural log of three ๐‘ก is five over ๐‘ก. d๐‘ฆ by d๐‘ฅ is the quotient of these. Itโ€™s d๐‘ฆ by d๐‘ก divided by d๐‘ฅ by d๐‘ก. So, thatโ€™s four plus five over ๐‘ก divided by five minus four over ๐‘ก.

Now, at the moment, this doesnโ€™t look particularly nice, so weโ€™re going to deal with the fractions in our numerator and denominator. Weโ€™re going to multiply through by ๐‘ก. Then, our numerator becomes four ๐‘ก plus five, and our denominator becomes five ๐‘ก minus four. And so, weโ€™re done. We found d๐‘ฆ by d๐‘ฅ. Itโ€™s four ๐‘ก plus five over five ๐‘ก minus four.

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