Derivatives of the Sine, Cosine and Tangent Functions

It can be shown from first principles that:

d(sin x)dx=cos x
d(cos x)dx=−sin x
d(tan x)dx=sec2x

In words, we would say:

The derivative of sin x is cos x,
The derivative of cos x is −sin x (note the negative sign!) and
The derivative of tan x is sec²x.

Now, if u = f(x) is a function of x, then by using the chain rule, we have:

d(sin u)dx=cos ududx
d(cos u)dx=−sin ududx
d(tan u)dx=sec2ududx

Example 1

y=sin(x2+3).

Differentiate y=sin(x2+3) First, let: 

u=x2+3 and so y=sin u.
We have:
dydx=dydududx=cos ududx=cos(x2+3)d(x2+3)dx=2x cos(x2+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.

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!

Example2

Find the derivative of 

In the final term, put u = 2x³.

We have:

y=3 sin 4x+5 cos 2x3

dydx=(3)(cos 4x)(4)+(5)(−sin 2x3)(6x2)=12 cos 4x−30x2

sin 2y=3 sin 4x+5 cos 2x3

=dydududx=−sin ududx=−sin(3x4)d(3x4)dx=−12x3sin 3x4

y=cos 3x4