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The Derivative Function
Derivatives of Polynomials
Power Rule
Quotient Rule
Chain Rule
100
Given: f(x)=1/(x-1) Find: f'(0)
-1
100
Apply differentiation rules to determine the derivative of the function: y = root(x)-3 root(x+1)
1/(2 root x) - 3/(2 root(x+1))
100
Determine f'(x) for: f(x) = (x + 1)(x
2
+ 1)(x
3
+ 1)
6x
5
+5x
4
+4x
3
+6x
2
+2x+1
100
Determine the slope of the tangent line at x = 7: f(x) = (x + x
-1
)(x + x
-2
)
1/2
100
Given f(x) = 3x + 1 and g(x) = x
2
-3 determine: g(f(x))
9x
2
+6x-2
200
Determine dy/dx from first principles (must show your full solution for points) y=(x+3)
2
1/(2 root(x-8))
200
Determine the equation of the tangent line at x = 1: f(x) = 7x
3
- 3x
-2
y = 27x - 23
200
Determine the equation of the tangent line at x = 1: f(x) = (3x
-3
- 3)(4x
-4
- 4)
y = 0
200
Determine the equation of a line that is normal to the tangent at x = 0: f(x) = 1/3 x
3
- 1/2 x
2
x = 0
200
Determine f'(x): f(x) = 1/(x
2
+ 1)
3
-6/(x
2
+ 1)
4
300
Determine the slope of a line perpendicular to the tangent line at the given x value. f(x) = 1/x at x = 3
9
300
Determine the slope of a line normal to the tangent line at x = -1. f(x) = -(x+7x
2
)
-1/13
300
Determine the slope of a line normal to the tangent line at x = -2
-1/26
300
Determine the equation of the normal to the graph of y = (x+1)(x
2
+ 4x + 1) at (0, 1)
x + 5y - 5 = 0
300
Differentiate f(x) = x
3
(2 - 5x
3
)
3
3x
2
(2 - 5x
3
)
3
+x
3
3(2 - 5x
3
)
2
(-15x
2
)
400
Determine the equation of a line that is normal to the tangent line at the given x-value: f(x) = x
2
+3x-6 at x = 2
y = -1/7x + 30/7
400
Determine the equation of a line that is normal to the tangent line at x = -1. f(x) = (2x
-1
- 7)
2
y = -1/36x + 2915/36
400
Determine the equation of a line that is perpendicular to the tangent line at x = 1
y = 1/4 x - 1/4
400
For what value(s) of x is the slope of the tangent line to y = (2x + 1)
3
negative?
none
400
Determine the slope of the tangent when x = 1: y = 1/(1 + x)
2
-1/4
500
Determine the value(s) of x for which each of the following functions has a horizontal tangent line. f(x) = x
3
- 3x
x = 1, x = -1
500
For what values of x do the following pair of graphs have tangent lines with the same slope? f(x) = x
2
- 3x g(x) = 1/x
x = -1/2 and x = 1
500
Determine the value(s) of x for which the function has a horizontal tangent line. f(x) = (x + 1)(x + 2)(x + 3)
x = -2 + root3/3 and x = -2 - root3/3
500
For what value(s) of x do the tangents to y = x
2
and y = x(x
2
- 1) have the same slope?
x = -1/3, x = 1
500
Determine the points on the graph of each function where the slope of the tangent is 0. f(x) = (1 - x)/x
2
(2, -1/4)