The derivative at a point
The definition of the derivative

When the limit can be found exactly
1.) Let f(x) = 3x^{2} +1. Using the definition of the derivative,
find the slope of the tanget to f(x)
where x = 2. Check your answer using the rule for derivatives of polynomials.
When the limit must be estimated
2.) Let f(x) = sin(2x). Using the definition of the derivative, find f ¢((p)/ 8). To evaluate the limit, use the estimation technique. Check your answer, using the chain rule and formula for the derivative of sin x.
When the limit doesn't exist
3.) Using the definition of the derivative show that the derivative of 2x + 1 does not exist for x = 1/2.
The derivative function
Using the definiton
Finding the derivative of a function given graphically
4.) Sketch the graph of the derivative of the function shown below.
Finding the derivative of a function given algebraically
5.) Let f(x) = ^{1}/_{x}. Using the definition of the derivative, find f ¢(x).
Interpreting the derivative as a rate of change
6.) A stone is dropped in a pool of water, creating circular ripples,
the first one growing larger as time passes. Let r = f(t) be the radius
(in inches) of the largest circle at time t seconds.
a.) What does it mean that f(2) = 8? Write a complete
sentence.
b.) What are the units of f ¢(t)?
c.) What does it mean that f ¢(2)
= 3? Write a complete sentence.
d.) What do you predict the value of f (3) is, given
that f(2)=8 and f '(2)=3?
The second derivative
Graphic implications of the second derivative  concavity
7.) Which of the points labeled by letters in the graph of f in the
following figure have:
a.) f positve? Do not include points where f = 0.
b.) f ¢ = 0?
c.) f ¢ positive?
Do not include points where f ¢ = 0.
d.) f ¢¢ positive?
Do not include points where f ¢¢ =
0.
e.) f ¢¢ negative?
Do not include points where f ¢¢ =
0.
Some rules for differentiation
Derivative of a constant
8.) What is the derivative of f(x) = p?
Derivative of a linear function
9.) What is the derivative of f(x) = x? What is the derivative of f(x) = 2x + 4?
Derivative of a constant times a function
Derivative of Sums and Differences
Power Rule
10.) What is the derivative of f(x) = x^{2}? What is the derivative of f(x) = x^{1/4}?
Derivative of a polynomial
11.) Find the equation of the tangent line to f(x) = x^{10}  9x^{7} + 24 x^{4}  3 x^{2} + ^{1}/_{2 , }at x = 1.
More Rules
The derivative of sinx
The derivative of cosx
The Chain Rule
12.) Find the derivative of sin(2x). Find the derivative of cos(x^{2} + p/ 2).
The derivative of lnx
13.) Find the derivative of sin( lnx).
The derivative of e^{x}
14.) Find the derivative of e^{1/x}.
The derivative of a^{x}
15.) Find the derivative of 1/ 2^{x}.