 At the beginning of chapter three in our lecture series, we first introduced the power rule and said that if you take any power function f of x equals x to the n, where n again is any real number, then the derivative of f of x will look like n times x theta minus one. At the time, we only proved it for the case of monomial functions, that is, situations where this number n was a positive integer and we claimed that it was true in general. We're now in that setting where we actually can prove it in general, that as we can prove the power rule for any real number and it follows as a consequence of logarithmic differentiation. So suppose we have our power function y equals x to the n. Well, we're gonna take the natural log of both sides, but the natural log of zero is undefined, so we are gonna assume that x is not zero for this argument. We'll treat zero separately in just a moment. So if we take the natural log of both sides, we'll get the natural log of the absolute value of y, which is equal to the natural log of the absolute value of x to the n right there. Well, by properties of the natural log, excuse me, of the absolute value, this becomes the natural log of the absolute value of x raised to the nth power. And then by properties of the logarithm, you can bring that exponent out in front as a coefficient and we get n times the natural log of the absolute value of x. This is equal to the natural log of the absolute value of y. So then let's take the derivative of both sides. Let's take the derivative with respect to x on both sides. The left-hand side will become y prime over y, like we always get when we do logarithmic differentiation. The right-hand side though, because we have this n times the natural log of the absolute value of x right here, we can factor out the n, it's a constant multiple. So we have to take the derivative of the natural log of the absolute value of x, but that's just equal to one over x. So we get n times one over x. Putting that together, we get this n over x right here. Now in order to get y prime, we have to solve for y. We're gonna times both sides of the equation by y so that they cancel out on the left-hand side. So that means that the derivative of y is equal n y over x, but y is just the power function into the x right there. Excuse me, x to the n. And so then plug that in for y, we get n times x to the n over x. And therefore, if we subtract the powers, the power will lower by one. And so we end up with n times x to the n minus one. And that's been the derivative. Isn't that pretty neat? We're able to get the power rule formula using logarithmic differentiation. We do have to treat the case where x equals zero separately, but we're gonna go back to the definition of the derivative in that situation. So if we have to compute f prime at zero, we're gonna take the limit of the difference quotient as h approaches infinity of zero plus h to the n minus zero to the n all over h. Well, if you take zero plus h, that's just an h. So you're gonna get h to the n. And if you take zero to any power, that's just gonna be zero. And so h to the n minus zero will just give you h to the n like we see here. But then if we have h to the n divided by h, by x what are rules, you can cancel out the h. And so you end up with an h to the n minus one, which as h approaches zero, this will then just become zero to the n minus one power, which is equal to zero. And anything times zero is just zero. So if you want to slap in some extra factor of n on there, and that'll then reproduce the formula as a special case when x equals zero. And so this shows us that the power rule is actually a special case of logarithmic differentiation. So we've been using the power rule for a good while here in chapter three unbeknownst to us, this is just a special case of a more advanced technique known as logarithmic differentiation.