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NARRATOR: Exercise three.

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Give an alternative proof

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that the derivative of log e of x
is one divided by x

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by differentiating
both sides of the equation

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X equals e
to the power of log e of x.

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So we start by differentiating
both sides of this equation.

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The derivative of x is 1.

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And the derivative of e
to the power of log e of x

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is going to require
the use of the chain rule.

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And the result of the chain rule
is e to the power of log e of x

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multiplied by
the derivative of log e of x.

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So as an aside,

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the chain rule involves us

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allowing y to equal

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e to the power of log e of x.

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So ultimately, we're trying to

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calculate the derivative of this,

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that is dy/dx.

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And we'll then let

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u equal log e of x.

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And this then implies that

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y is equal to e to the power of u.

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And so the derivative dy/du

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is also e to the power of u.

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The chain rule states that

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dy/dx is equal to

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dy/du multiplied by du/dx.

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And we've calculated dy/du

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to be e to the power of u.

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So substituting this in,

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we also know that u is log e of x.

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So e to the power of u is in fact

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e to the power of log e of x

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and du/dx will be the derivative

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of log e of x,

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which is what we're trying to prove.

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So we'll leave this written

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as the derivative of log e of x.

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And so we get the equation that

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1 is equal to

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e to the power of log e of x

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multiplied by
the derivative of log e of x.

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So e to the power of log e of x
is equal to x.

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And then simply dividing
both sides of the equation by x

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we get the final result that
the derivative of log e of x

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is in fact 1 divided by x.