Show:
Basic Derivatives
Product and Quotient Rule
Chain Rule
Implicit Differentiation
The Derivative
100
What is the derivative of y = 3x2 ?
y' = 6x
100
Find the derivative of y = 3x2 sin x
3x2 cos x + 6x sin x
100
Find y' if y = (3x - 9)4
12(3x - 9)3
100
Find the derivative implicitly x2 + y2 = 30
y' = - x/y
100

What does the derivative represent?

The slope of a tangent line

200
Find the derivative of f(x) = 5
0
200
f(x) = (3x - 2x2)(5 + 4x)
-24x2 + 4x + 15
200
Find the derivative of f(x) = sin 4x
4 cos 4x
200
sin x + 2cos 2y = 1
y' = cosx/4sin2y
200

True or false:  Derivatives are defined as a limit

True

300
Find the derivative of f(x) = 6x3 + x - 2
18x2 + 1
300
Find the derivative of y = (5x - 2)/(x2 + 1)
(-5x2 + 4x + 5)/(x2 + 1)2
300
Find y' if y = (9x2 + 4)1/3
(6x)/(9x2 + 4)2/3
300
(y - 2)2 = 4x - 12
y' = 2/(y - 2)
300

If a function is continuous over its entire domain, what does this mean for the derivative?

The derivative must also exist over the entire domain.

400
Find the derivative of f(x) = 3sinx - 2cosx
3cosx + 2sinx
400
Find the derivative of f(x) = (2x + 5)/(x)1/2
(2x - 5)/(2x3/2)
400
f(x) = sin3 4x Find f'(x)
12sin2 4x cos 4x
400
x3 - 3x2y + 2xy2 = 12
y' = (-3x2 + 6xy - 2y2)/(-3x2 + 4xy)
400

If a ball's height in the air is given by the equation h(t) = -4.9t2 + 5t + 4.5, find the time at which the ball reaches its highest point. 

In order to solve this problem, consider what a derivative represents.  How could we use the graph of h(t) to tell where the ball is at its highest point?

answer: 0.51 seconds

500

Differentiate:

f(x) = 3x^2  +  (2/3)x^-3  -  x^-1

f'(x) = -2x^-4  +  x^-2  +  6x

500

Find the derivative:

f(x) = (x sin(x)) / (2x - 1)

f'(x) = [-sin(x) + 2x^2 cos(x) - xcos(x)] / (2x-1)^2

500
Find the derivative of f(2):


y = [x/(x+1)]^3

f'(2) = 4/27

500

Find the derivative:

ln(xy) = 2x

y'= (2xy - y) / x

500

Give the equation for the limit definition of the derivative.

lim(h->0) [f(x+h) - f(x)] / h