WEBVTT

1
00:00:00.000 --> 00:00:06.188
Here is the problem: derivative of (arctan(x squared plus 1)) plus (arctan(x squared plus 1)).

2
00:00:06.638 --> 00:00:13.279
Find the derivative of (atan(x squared plus 1)) plus (atan(x squared plus 1)) with respect to x.

3
00:00:14.079 --> 00:00:17.202
Sum rule: differentiate each term separately.

4
00:00:17.652 --> 00:00:23.689
Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared).

5
00:00:24.139 --> 00:00:27.401
Sum rule: differentiate each term separately.

6
00:00:27.851 --> 00:00:34.457
Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

7
00:00:35.257 --> 00:00:36.859
Combine and simplify.

8
00:00:37.659 --> 00:00:44.114
So the answer is f prime of x equals 4x / ((x squared plus 1) squared plus 1).

9
00:00:44.564 --> 00:00:57.718
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).

10
00:00:58.518 --> 00:01:05.298
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.