WEBVTT

1
00:00:00.000 --> 00:00:03.971
Here is the problem: d by d x sin(3x).

2
00:00:04.421 --> 00:00:08.008
Find the derivative of sin(3 x) with respect to x.

3
00:00:08.808 --> 00:00:13.626
Chain rule — d by d x[sin u] equals cos u times u prime.

4
00:00:14.076 --> 00:00:15.550
Constant multiple rule.

5
00:00:16.000 --> 00:00:17.532
Combine and simplify.

6
00:00:18.333 --> 00:00:22.594
So the answer is f prime of x equals 3cos(3x).

7
00:00:23.044 --> 00:00:36.210
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
00:00:37.010 --> 00:00:43.964
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.