WEBVTT

1
00:00:00.000 --> 00:00:03.785
Here is the problem: differentiate x to the power x.

2
00:00:04.235 --> 00:00:08.020
Find the derivative of x to the power x with respect to x.

3
00:00:08.820 --> 00:00:19.304
Logarithmic differentiation: d by d x[uᵛ] equals uᵛ(v prime ln u plus v u prime /u).

4
00:00:19.754 --> 00:00:21.345
Combine and simplify.

5
00:00:22.145 --> 00:00:27.683
So the answer is f prime of x equals (x) to the power x times (ln(x) plus 1).

6
00:00:28.133 --> 00:00:40.788
The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

7
00:00:41.588 --> 00:00:48.229
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.