# 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 โ 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๐ฅ, where ๐ 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.