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Let π of π₯ be equal to arcsin of π₯ to the power of four cubed.
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Find π prime of π₯.
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π of π₯ is expressed as a function of a function of a function.
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Itβs a composite function.
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Weβre therefore going to need to use the chain rule to find the derivative π prime of π₯.
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The chain rule says that if π¦ is a function of π’ and π’ is a function of π₯, then dπ¦ by dπ₯ is equal to dπ¦ by dπ’ times dπ’ by dπ₯.
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A special case of the chain rule is the general power rule.
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And this says that if π’ is a function of π₯, then the derivative of π’ to the power of π can be written as π times π’ to the power of π minus one multiplied by the derivative of π’ with respect to π₯.
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Weβre actually going to apply both of these rules during this question.
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We can use the general power rule to begin finding π prime of π₯.
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Since our function in π₯ is arcsin of π₯ to the power of four and then thatβs being cubed, we can say that π prime of π₯ must be equal to three times that function arcsin of π₯ to the power of four, and then thatβs squared.
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And we multiply that by the derivative of arcsin of π₯ to the power of four with respect to π₯.
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So weβre going to need to use the chain rule to actually evaluate the derivative of arcsin of π₯ to the power of four with respect to π₯.
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Weβll say that π¦ is equal to arcsin of π’ and π’ is equal to π₯ to the power of four.
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To use the chain rule, weβre going to need to find the derivative of each of these.
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The derivative of π’ with respect to π₯ is four π₯ cubed.
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Weβll also use the fact that the derivative of arcsin of π₯ with respect to π₯ is one over the square root of one minus π₯ squared.
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This means that dπ¦ by dπ’ is one over the square root of one minus π’ squared.
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We substitute this back into the formula for the chain rule.
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And we see that the derivative of arcsin of π₯ to the power of four with respect to π₯ is one over the square root of one minus π’ squared times four π₯ cubed.
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Remember that weβre trying to differentiate this with respect to π₯.
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So weβre going to use the fact that we let π’ be equal to π₯ to the power of four.
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And when we substitute this back into the expression for the derivative, we get four π₯ cubed over the square root of one minus π₯ to the power of four all squared.
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Now, in fact, π₯ to the power of four squared is π₯ to the power of eight.
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And we can replace this in our original equation for π prime of π₯.
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And we get three arcsin of π₯ to the power of four squared times four π₯ cubed over the square root of one minus π₯ to the power of eight.
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Simplifying just a little, and we find that π prime of π₯, the derivative of our function π with respect to π₯, is 12 arcsin of π₯ to the power of four squared over the square root of one minus π₯ to the power of eight.