Text description - screencast for exercise 3

[The narrator reads out the onscreen text.]

NARRATOR: Exercise three. Give an alternative proof that the derivative of log e of x is one divided by x by differentiating both sides of the equation X equals e to the power of log e of x. So we start by differentiating both sides of this equation. The derivative of x is 1. And the derivative of e to the power of log e of x is going to require the use of the chain rule. And the result of the chain rule is e to the power of log e of x multiplied by the derivative of log e of x.

So as an aside, the chain rule involves us allowing y to equal e to the power of log e of x. So ultimately, we're trying to calculate the derivative of this, that is dy/dx. And we'll then let u equal log e of x. And this then implies that y is equal to e to the power of u. And so the derivative dy/du is also e to the power of u. The chain rule states that dy/dx is equal to dy/du multiplied by du/dx. And we've calculated dy/du to be e to the power of u. So substituting this in, we also know that u is log e of x. So e to the power of u is in fact e to the power of log e of x and du/dx will be the derivative of log e of x, which is what we're trying to prove.

So we'll leave this written as the derivative of log e of x. And so we get the equation that 1 is equal to e to the power of log e of x multiplied by the derivative of log e of x. So e to the power of log e of x is equal to x. And then simply dividing both sides of the equation by x we get the final result that the derivative of log e of x is in fact 1 divided by x.