Here are two limits that are important, but the proofs are long and tedious. Therefore, I will state them without proof.

FACT:

FACT:

EXAMPLE 1:

SOLUTION:

EXAMPLE 2:

SOLUTION:

EXAMPLE 3:

SOLUTION:

EXAMPLE 4:

SOLUTION:

EXAMPLE 5:

SOLUTION:

Now, I will state the derivatives of the trigonometric functions.

FACT: 

FACT:

FACT:

FACT:

FACT:

FACT:

EXAMPLE 6:

Find the first derivative of y = -10x + 3cos x.

SOLUTION: y' = -10 + 3(-sin x) = -10 -3sin x

EXAMPLE 7:

Find the first derivative of y = x ² tan x - x ^{- 2}.

SOLUTION: For the first part of this derivative, I will have to use the product rule.

y' = (2x) tan x + x ² (sec ² x) - (-2x ^{- 3}) = 2x tan x + x ² sec ² x + 2x ^{- 3}

EXAMPLE 8:

SOLUTION:

EXAMPLE 9:

SOLUTION:

EXAMPLE 10:

Find the first derivative of y = x ² - sec x + 1.

SOLUTION:

y' = 2x - (sec x)(tan x) + 0 = 2x - (sec x)(tan x)

EXAMPLE 11:

SOLUTION:

EXAMPLE 12:

Graph y = 1 + cos x over the interval -3p /2 £ x £ 2p , together with its tangent lines at the points x = - p /3 and x = 3p /2.

SOLUTION: To find the tangent lines, I must find the first derivative of the function. From the first derivative, I can find the slopes of the tangent lines.

y' = - sin x

Now to find the other tangent line.

Here is the graph of the original function and the two tangent lines. y = 1 + cos x is in blue. The tangent line at the point where x = -p /3 is in red, and the tangent line at the point x = 3p /2 is in green.