WEBVTT

1
00:00:00.000 --> 00:00:05.457
Here is the problem: derivative of ((2x) cubed) plus (ln(x squared plus 1)).

2
00:00:05.907 --> 00:00:12.246
Find the derivative of ((2x) cubed) plus (ln(x squared plus 1)) with respect to x.

3
00:00:13.046 --> 00:00:16.134
Sum rule: differentiate each term separately.

4
00:00:16.584 --> 00:00:21.727
Chain rule — d by d x[ln u] equals u prime /u.

5
00:00:22.177 --> 00:00:25.312
Sum rule: differentiate each term separately.

6
00:00:25.762 --> 00:00:32.554
Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

7
00:00:33.354 --> 00:00:41.609
Power rule with the chain rule: d by d x[u to the power n] equals n times u to the power n minus 1 times u prime.

8
00:00:42.059 --> 00:00:43.533
Constant multiple rule.

9
00:00:43.983 --> 00:00:45.527
Combine and simplify.

10
00:00:46.327 --> 00:00:53.931
So the answer is f prime of x equals 24x squared plus 2x / (x squared plus 1).

11
00:00:54.381 --> 00:01:07.361
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).

12
00:01:08.161 --> 00:01:14.686
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.