WEBVTT

1
00:00:00.000 --> 00:00:03.959
Here is the problem: derivative of x/(x plus 1).

2
00:00:04.409 --> 00:00:08.925
Find the derivative of x / (x plus 1) with respect to x.

3
00:00:09.725 --> 00:00:16.645
Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared.

4
00:00:17.095 --> 00:00:20.195
Sum rule: differentiate each term separately.

5
00:00:20.645 --> 00:00:22.212
Combine and simplify.

6
00:00:23.012 --> 00:00:27.819
So the answer is f prime of x equals 1 / (x plus 1) squared.

7
00:00:28.268 --> 00:00:40.946
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).

8
00:00:41.746 --> 00:00:48.503
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.