WEBVTT
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Evaluate π prime of one, where π of π₯ is equal to one minus six divided by three π₯ minus five.
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The question gives us a function π of π₯, and it wants us to find π prime of one.
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Thatβs the first derivative of π evaluated at π₯ is equal to one.
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So letβs start by finding an expression for π prime of π₯.
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Thatβs the derivative of one minus six divided by three π₯ minus five with respect to π₯.
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We can simplify this by using the fact that the derivative of the difference between two functions is equal to the difference of their derivatives.
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This gives us the derivative of one with respect to π₯ minus the derivative of six divided by three π₯ minus five with respect to π₯.
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However, one is just a constant, so its derivative is equal zero.
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To evaluate our second derivative, we see itβs the quotient to two functions.
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So weβll do this by using the quotient rule.
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The quotient rule tells us for functions π’ and π£, the derivative of π’ divided by π£ is equal to π£π’ prime minus π’π£ prime all divided by π£ squared.
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Weβll set π’ to be the function in our numerator.
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Thatβs six.
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And weβll set π£ equal to the function in our denominator.
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Thatβs three π₯ minus five.
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We now want to find expressions for π’ prime and π£ prime.
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First, we see that π’ is a constant six.
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So its derivative is equal to zero.
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Next, to differentiate three π₯ minus five, we recall the power rule for differentiation.
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To differentiate ππ₯ to the πth power, we multiply by the exponent and reduce the exponent by one.
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This tells us the derivative of three π₯ is equal to three.
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And we know the derivative of the constant negative five is equal to zero.
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Weβre now ready to find the derivative of this quotient by using the quotient rule.
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And remember, we need to multiply this value by negative one.
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Substituting our values for π’, π£, π’ prime, and π£ prime, we get negative one times three π₯ minus five times zero minus six times three all divided by three π₯ minus five squared.
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And we can simplify this expression.
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Three π₯ minus five times zero is equal to zero.
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And then negative one times negative six times three is equal to 18.
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So weβve found an expression for π prime of π₯.
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Itβs equal to 18 divided by three π₯ minus five squared.
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But remember, the question wants us to find the value of π prime of π₯ when π₯ is equal to one.
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So we need to substitute π₯ is equal to one into our expression for π prime of π₯.
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Doing this, we get π prime of one is equal to 18 divided by three times one minus five squared.
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And finally, we can just evaluate this expression.
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We have three times one minus five is equal to negative two.
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So we get 18 divided by negative two squared.
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And negative two squared is equal to four.
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Finally, we simplify 18 divided by four to get nine divided by two.
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Therefore, weβve shown if π of π₯ is equal to one minus six divided by three π₯ minus five, then π prime of π₯ evaluated at π₯ is equal to one is equal to nine divided by two.