WEBVTT

1
00:00:00.000 --> 00:00:04.772
Here is the problem: derivative of (sin(x cubed))((2x) cubed).

2
00:00:05.222 --> 00:00:09.796
Find the derivative of (sin(x cubed)) ((2x) cubed) with respect to x.

3
00:00:10.596 --> 00:00:15.275
Product rule: (uv) prime equals u prime v plus uv prime.

4
00:00:15.725 --> 00:00:24.282
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.

5
00:00:24.732 --> 00:00:26.172
Constant multiple rule.

6
00:00:26.622 --> 00:00:31.579
Chain rule — d by d x[sin u] equals cos u times u prime.

7
00:00:32.379 --> 00:00:39.078
Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

8
00:00:39.528 --> 00:00:41.119
Combine and simplify.

9
00:00:41.919 --> 00:00:50.963
So the answer is f prime of x equals 24(x to the power 5 times cos(x cubed) plus sin(x cubed) times x squared).

10
00:00:51.413 --> 00:01:04.416
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).

11
00:01:05.216 --> 00:01:11.880
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.