WEBVTT

1
00:00:00.000 --> 00:00:06.246
Here is the problem: derivative of (e to the power 2x) plus (ln(3x minus 2)).

2
00:00:06.696 --> 00:00:12.896
Find the derivative of (e to the power 2x) plus (ln(3 x minus 2)) with respect to x.

3
00:00:13.696 --> 00:00:16.773
Sum rule: differentiate each term separately.

4
00:00:17.223 --> 00:00:22.355
Chain rule — d by d x[ln u] equals u prime /u.

5
00:00:22.805 --> 00:00:26.207
Difference rule: differentiate each term separately.

6
00:00:26.657 --> 00:00:28.143
Constant multiple rule.

7
00:00:28.943 --> 00:00:36.571
Exponential rule: d by d x[eᵘ] equals eᵘ times u prime.

8
00:00:37.021 --> 00:00:38.507
Constant multiple rule.

9
00:00:38.957 --> 00:00:40.536
Combine and simplify.

10
00:00:41.336 --> 00:00:48.906
So the answer is f prime of x equals 2e to the power 2x plus 3 / (3 x minus 2).

11
00:00:49.356 --> 00:01:02.406
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
00:01:03.206 --> 00:01:09.940
Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.