On-line Math 21

3.1  Trigonometric functions

3.1.3  Derivatives of sin(x) and cos(x) .

The derivative of sin(x) .

The derivative of the sine function sin(x) is the first derivative you can't figure out from the definition just using enough algebra to cancel a power of h top and bottom. However, what it does take is the trigonometric limits we just derived.

Theorem 1 The functions sin(x) and cos(x) have the following derivatives:

(sin(x))¢ = cos(x) (cos(x))¢ = -sin(x)

  Proof:

Example 1 Find (tan(x))¢.

Solution

Example 2 Find (sec(x))¢.

Solution

Exercise 1 Find (cot(x))¢ =

This is similar to the previous ones. Watch for signs ( ±).

Exercise 2 Find (csc(x))¢ =

Example 3 Find (x³cos(x))¢.

Solution

Exercise 3 Find (sin(x)cos(x))¢ =

Example 4 Find the tangent line to the curve y = sin(x) at (p/6,1/2) , and use that to find, approximately, sin(p/6+0.1) .

Solution