Derivative Rules

The Derivative Function

100

Given: f(x)=1/(x-1) Find: f'(0)

-1

200

Determine dy/dx from first principles (must show your full solution for points) y=(x+3)²

1/(2 root(x-8))

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

400

Determine the equation of a line that is normal to the tangent line at the given x-value: f(x) = x²+3x-6 at x = 2

y = -1/7x + 30/7

500

Determine the value(s) of x for which each of the following functions has a horizontal tangent line. f(x) = x³ - 3x

x = 1, x = -1

Derivatives of Polynomials

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))

200

Determine the equation of the tangent line at x = 1: f(x) = 7x³ - 3x^-2

y = 27x - 23

300

Determine the slope of a line normal to the tangent line at x = -1. f(x) = -(x+7x²)

-1/13

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

500

For what values of x do the following pair of graphs have tangent lines with the same slope? f(x) = x² - 3x g(x) = 1/x

x = -1/2 and x = 1

Product Rule

100

Determine f'(x) for: f(x) = (x + 1)(x² + 1)(x³ + 1)

6x⁵+5x⁴+4x³+6x²+2x+1

200

Determine the equation of the tangent line at x = 1: f(x) = (3x^-3 - 3)(4x^-4 - 4)

y = 0

300

Determine the slope of a line normal to the tangent line at x = 4 of f(x)=(1/x-1)root x

16/5

400

Determine the equation of a line that is perpendicular to the tangent line at x = 1 of f(x)=

y = 1/4 x - 1/4

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

Quotient Rule

100

Determine the slope of the tangent line at x = 7: f(x) = (x + x^-1)(x + x^-2)

1/2

200

Determine the equation of a line that is normal to the tangent at x = 0: f(x) = 1/3 x³ - 1/2 x²

x = 0

300

Determine the equation of the normal to the graph of y = (x+1)(x² + 4x + 1) at (0, 1)

x + 5y - 5 = 0

400

For what value(s) of x is the slope of the tangent line to y = (2x + 1)³ negative?

none

500

For what value(s) of x do the tangents to y = x² and y = x(x² - 1) have the same slope?

x = -1/3, x = 1

Chain Rule

100

Given f(x) = 3x + 1 and g(x) = x² -3 determine: g(f(x))

9x²+6x-2

200

Determine f'(x): f(x) = 1/(x² + 1)³

-6/(x² + 1)⁴

300

Differentiate f(x) = x³(2 - 5x³)³

3x²(2 - 5x³)³+x³3(2 - 5x³)²(-15x²)

400

Determine the slope of the tangent when x = 1: y = 1/(1 + x)²

-1/4

500

Determine the points on the graph of each function where the slope of the tangent is 0. f(x) = (1 - x)/x²

(2, -1/4)