First, let: `u = x^2+ 3` and so `y = sin u`.

We have:

`(dy)/(dx)=(dy)/(du)(du)/(dx)`

`=cos u(du)/(dx)`

`=cos(x^2+3)(d(x^2+3))/(dx)`

`=2x\ cos(x^2+3)`

IMPORTANT:

cos x2 + 3

does not equal

cos(x2 + 3).

The brackets make a big difference. Many students have trouble with this.

Here are the graphs of y = cos x2 + 3 (in green) and y = cos(x2 + 3) (shown in blue).

The first one, y = cos x2 + 3, or y = (cos x2) + 3, means take the curve y = cos x2 and move it up by `3` units.

Graph y = cos(x^2+3)

The second one, y = cos(x2 + 3), means find the value (x2 + 3) first, then find the cosine of the result.

They are quite different!

Graph y = cos(x^2+3)