WEBVTT
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Determine the third derivative of the function π¦ is equal to negative six times the cos of two π₯.
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Weβre given π¦ which is a trigonometric function in π₯.
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And weβre asked to find the third derivative of π¦.
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This will be the third derivative of π¦ with respect to π₯.
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So weβll start by finding the first derivative of π¦ with respect to π₯.
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Thatβs the derivative of negative six cos of two π₯ with respect to π₯.
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And to do this, we can recall one of our standard trigonometric derivative results.
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For any real constant π, the derivative of negative cos of ππ₯ with respect to π₯ is equal to π times the sin of ππ₯.
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And we can see our derivative is not exactly in this form.
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Weβll take the constant factor of six outside of our derivative.
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So this gives us the following expression, and we can see this is exactly in this form with our value of π set to be equal to two.
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So by setting our value of π equal to two, we can differentiate negative the cos of two π₯ with respect to π₯ to get two times the sin of two π₯.
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Then we can simplify our coefficient in this expression.
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Six times two is equal to 12.
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So weβve shown dπ¦ by dπ₯ is equal to 12 sin of two π₯.
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But remember, we need to find an expression for the third derivative of π¦ with respect to π₯.
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We need to differentiate this again.
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We differentiate to find d two π¦ by dπ₯ squared is equal to the derivative of 12 sin of two π₯ with respect to π₯.
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And once again, weβll simplify this expression.
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Weβll take the constant factor of 12 outside of our derivative.
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So to find our expression for d two π¦ by dπ₯ squared, we need to differentiate the sin of two π₯ with respect to π₯.
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And once again, we can do this by using a standard trigonometric derivative result.
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For any real constant π, the derivative of the sin of ππ₯ with respect to π₯ is equal to π times the cos of ππ₯.
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Once again, we can see the value of our constant π is equal to two.
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So by using π is equal to two, we get d two π¦ by dπ₯ squared is equal to 12 times two cos of two π₯.
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And we can simplify our coefficient.
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12 times two is equal to 24.
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So weβve shown the second derivative of π¦ with respect to π₯ is equal to 24 times the cos of two π₯.
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To find our third derivative of π¦ with respect to π₯, we just need to differentiate this with respect to π₯ one more time.
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So we need to differentiate 24 cos of two π₯ with respect to π₯.
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And we can do this by using a trigonometric derivative result.
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For any real constant π, the derivative of the cos of ππ₯ with respect to π₯ is equal to negative π times the sin of ππ₯.
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Just as we did before, weβll take the constant factor of 24 outside of our derivative and then evaluate our derivative by setting the value of π equal to two.
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Doing this gives us d three π¦ by dπ₯ cubed is equal to 24 times negative two sin of two π₯.
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And finally, by simplifying our coefficient, we get negative 48 times the sin of two π₯, which is our final answer.
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Therefore, we were able to show if π¦ is equal to negative six times the cos of two π₯, then the third derivative of π¦ with respect to π₯ will be equal to negative 48 times the sin of two π₯.