00:01
So we want to use logarithmic differentiation to find the derivative of y is equal to x plus 1 to the x.
00:12
So let's go ahead and just first take natural logs of everything.
00:17
So natural log here, natural log here.
00:23
So we get y there and then x plus 1 to the x.
00:27
Now remember one of the rules for natural logs is if we have something to a power, we can move that power out front.
00:33
So we can rewrite this as natural log of y is equal to x times the natural log of x plus 1.
00:43
Now let's go ahead and take the derivative of this with respect to x.
00:47
So d by d x.
00:51
So the left -hand side will become 1 over y because that's the derivative of natural log.
00:58
And then we multiply by y prime since we are using implicit differentiation.
01:04
All right.
01:07
Now to take the derivative of x times natural log of x plus 1, we need to use product rule.
01:12
So product rule says we first take our first term, multiply it by the derivative of the second, d by d x, of natural log of x plus 1, then we switch the order of them.
01:27
So natural log of x plus 1, derivative with respect to x of x...