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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.
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In this question, we have a parametric equation where the coordinates π₯ and π¦ are given in terms of π‘.
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We recall that dπ¦ by dπ₯ is equal to dπ¦ by dπ‘ divided by dπ₯ by dπ‘.
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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‘.
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Letβs begin by considering our π₯-coordinate, π‘ minus two multiplied by four π‘ plus three.
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We can distribute the parentheses or expand the brackets here by using the FOIL method.
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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‘.
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Finally, multiplying the last terms gives us negative six.
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This can be simplified so that π₯ is equal to four π‘ squared minus five π‘ minus six.
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We can work out an expression for dπ₯ by dπ‘ by differentiating this with respect to π‘ term by term.
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Differentiating four π‘ squared gives us eight π‘, differentiating negative five π‘ gives us negative five, and differentiating any constant gives us zero.
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Therefore, dπ₯ by dπ‘ is equal to eight π‘ minus five.
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We can then repeat this process for our π¦-coordinate.
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We know that π¦ is equal to three π‘ squared minus four multiplied by π‘ minus three.
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Distributing the parentheses here gives us π¦ is equal to three π‘ cubed minus nine π‘ squared minus four π‘ plus 12.
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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.
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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.
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We want to calculate this when π‘ is equal to zero.
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Nine π‘ squared, 18π‘, and eight π‘ will all be equal to zero.
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This means that dπ¦ by dπ₯ is equal to negative four over negative five.
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Dividing a negative number by a negative number gives us a positive answer.
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Therefore, dπ¦ by dπ₯ is equal to four-fifths.