WEBVTT
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here we have a composite function and we're going to
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identify the inside function and the outside function before we
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differentiate. So the inside function would be the square
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root of X that is inside the E to the
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X function. And I would like to write the
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square root of X as extra the 1/2 when I
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differentiate it. So then the outside function is f
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of X equals e to the X okay, to
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find the derivative Do I d. X? What
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we want to dio according to the chain rule is
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start by taking the derivative of the outside function and
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we have learned that the derivative of each of the
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X is e to the X, So the derivative
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of each of the square root X we're going to
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have to start with the to the square root X
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. Then we multiply by the derivative of the inside
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function. So now we're taking the derivative of X
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to the 1/2 power and that would be 1/2 times
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X to the negative 1/2 power just using the power
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rule. Okay, so we have our derivative and
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now we're going to simplify So one of the things
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we can do is eliminate the negative exponents. So
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we have e to the square root X times 1/2
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x to the 1/2 because X to the negative 1/2
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is equivalent to one over X to the 1/2 and
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now we can just write it as a single fraction
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. So we have e to the square root X
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over two X to the 1/2. But why don't
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we take it one step further and change back the
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X to the 1/2 into radical notation and we have
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each of the square root x over two square root
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X that's are derivative.