Video Transcript
Given that ๐ฅ equals five ๐ก minus four times the natural log of ๐ก and ๐ฆ equals four ๐ก plus five times the natural log of three ๐ก, find d๐ฆ by d๐ฅ.
Here weโve been given a pair of parametric equations. These are equations for ๐ฅ and ๐ฆ in terms of a third parameter. Thatโs ๐ก. And weโre looking to find the derivative of ๐ฆ with respect to ๐ฅ. And so, we recall that as long as d๐ฅ by d๐ก is not equal to zero, the derivative of ๐ฆ with respect to ๐ฅ is equal to d๐ฆ by d๐ก divided by d๐ฅ by d๐ก. And so, it should be quite clear to us that weโre going to need to differentiate each of our respective equations with respect to ๐ก. And weโll do this term by term.
The first term in our equation for ๐ฅ is five ๐ก. Now, the derivative of five ๐ก with respect to ๐ก is five. Next, we recall that the derivative of the natural log of ๐๐ฅ, where ๐ is some constant, with respect to ๐ฅ is one over ๐ฅ. So, the derivative of the natural log of ๐ก is one over ๐ก. And so, the derivative of negative four times the natural log of ๐ก is negative four over ๐ก.
Letโs repeat this process for d๐ฆ by d๐ก. The derivative of four ๐ก with respect to ๐ก is four. Then, the derivative of the natural log of three ๐ก must be one over ๐ก. So, five times the natural log of three ๐ก is five over ๐ก. d๐ฆ by d๐ฅ is the quotient of these. Itโs d๐ฆ by d๐ก divided by d๐ฅ by d๐ก. So, thatโs four plus five over ๐ก divided by five minus four over ๐ก.
Now, at the moment, this doesnโt look particularly nice, so weโre going to deal with the fractions in our numerator and denominator. Weโre going to multiply through by ๐ก. Then, our numerator becomes four ๐ก plus five, and our denominator becomes five ๐ก minus four. And so, weโre done. We found d๐ฆ by d๐ฅ. Itโs four ๐ก plus five over five ๐ก minus four.