WEBVTT

1
00:00:00.000 --> 00:00:02.322
Here is the problem: sin(x).

2
00:00:02.772 --> 00:00:06.174
Find the second derivative of sin(x) with respect to x.

3
00:00:06.974 --> 00:00:08.286
Derivative number 1.

4
00:00:08.736 --> 00:00:11.813
d by d x[sin x] equals cos x.

5
00:00:12.263 --> 00:00:14.202
Simplify the first derivative.

6
00:00:14.652 --> 00:00:16.045
Derivative number 2.

7
00:00:16.845 --> 00:00:20.015
d by d x[cos x] equals negative sin x.

8
00:00:20.465 --> 00:00:22.334
Simplify the second derivative.

9
00:00:23.134 --> 00:00:27.337
So the answer is f prime prime (x) equals negative sin(x).

10
00:00:27.787 --> 00:00:40.755
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
00:00:41.555 --> 00:00:48.254
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.