### Video Transcript

Determine dπ¦ by dπ₯ at π‘ equals zero, given that π₯ is equal to π‘ minus two multiplied by four π‘ plus three and π¦ is equal to three π‘ squared minus four multiplied by π‘ minus three.

In this question, we have a parametric equation where the coordinates π₯ and π¦ are given in terms of π‘. We recall that dπ¦ by dπ₯ is equal to dπ¦ by dπ‘ divided by dπ₯ by dπ‘. This can also be written as dπ¦ by dπ‘ multiplied by dπ‘ by dπ₯, where dπ‘ by dπ₯ is the reciprocal of dπ₯ by dπ‘. Letβs begin by considering our π₯-coordinate, π‘ minus two multiplied by four π‘ plus three. We can distribute the parentheses or expand the brackets here by using the FOIL method.

Multiplying the first terms gives us four π‘ squared, multiplying the outside terms gives us three π‘, and multiplying the inside terms gives us negative eight π‘. Finally, multiplying the last terms gives us negative six. This can be simplified so that π₯ is equal to four π‘ squared minus five π‘ minus six. We can work out an expression for dπ₯ by dπ‘ by differentiating this with respect to π‘ term by term. Differentiating four π‘ squared gives us eight π‘, differentiating negative five π‘ gives us negative five, and differentiating any constant gives us zero. Therefore, dπ₯ by dπ‘ is equal to eight π‘ minus five.

We can then repeat this process for our π¦-coordinate. We know that π¦ is equal to three π‘ squared minus four multiplied by π‘ minus three. Distributing the parentheses here gives us π¦ is equal to three π‘ cubed minus nine π‘ squared minus four π‘ plus 12. Once again, we can differentiate this term by term with respect to π‘. dπ¦ by dπ‘ is equal to nine π‘ squared minus 18π‘ minus four.

As dπ¦ by dπ₯ is equal to dπ¦ by dπ‘ divided by dπ₯ by dπ‘, this is equal to nine π‘ squared minus 18π‘ minus four divided by eight π‘ minus five. We want to calculate this when π‘ is equal to zero. Nine π‘ squared, 18π‘, and eight π‘ will all be equal to zero. This means that dπ¦ by dπ₯ is equal to negative four over negative five. Dividing a negative number by a negative number gives us a positive answer. Therefore, dπ¦ by dπ₯ is equal to four-fifths.