Derivatives

Simple Dervatives

100

What is the derivative of 4x⁷

28x⁶

200

What is the derivative 3x⁴tan(x)?

3(4x³tan(x)+x⁴sec(x))

300

What is the derivative of (3x-9/2-x)?

15/(2-x)²

400

What is the derivative of 5e^x/3e^x+1?

5e^x/(3e^x+1)2

500

What is the derivative of 2^x/cos⁵(5x)?

2^x(ln(2)cos(5x)+20sin(5x)) / cos⁶(5x)

Implicit Differentiation

100

Find dy/dx by implicit differentiation for x/y³=1

y/3x

200

Find dy/dx by implicit differentiation for 7y²+sin(3x)=12−y⁴

dy/dx = -3cos(3x) / 14y+4y³

300

Find dy/dx by implicit differentiation for e^xy=e^4x-e^5y

(4e^4x-ye^xy) / (xe^xy+5e^5y)

400

Find d²y/dx² by implicit differentiation for 2x²-3y²=4

2(3y²-x²)/9y³

500

Find d²y/dx² by implicit differentiation for xy + y²=2

2y(x+y) = (x+2y)³

Derivatives lines

100

What is the equation of the tangent line to y=x² at x=2

y=4x-4

200

What is the equation of the tangent line to y = -4x³+7x²-9x+12 at (1,6)

y=-7x+13

300

If y=16x^-1-x². At what point is the tangent line horizontal?

(-2,-12)

400

What is the equation of the tangent line to y²e^2x=3y+x² at (0,3)

y=-6x+3

500

A line normal (perpendicular) to the curve y=2x² at a point in the first quadrant also passes through the point (0,3/4). Find an equation for this line

y=(-1/2)x+3/4

Derivative Applications

100

Find the point of inflection for f(x) = x³-6x²+12x-5

x = 2

200

Let f(x) = 5x³ + 2x² - 3x. Find interval of concave down.

(-infinity, -2/15)

300

Find the point(s) of inflection: f(x) = -7x⁴+5x²-x+3

(You may need a calculator)

(-sqrt(5/42), sqrt(5/42) + 881/252), (sqrt(5/42), -sqrt(5/42)+ 881/252)

400

Let f(x) = x⁶/30 - x⁵/20 - x⁴ + 3x + 20. Find the intervals of concave up.

(-infinity, -3) U (4 , infinity)

500

Find and identify the relative extrema for f(x) = 5x⁵ - 7x⁴ + 8x³ - x + 8

(-0.1827, 8.125) (relative max)

(0.2301, 7.851) (relative min)

Velocity Questions

100

A particle moving right along the x axis has a velocity modeled by the function v(t)= lnx +e^x. Find the acceleration function

1/x+e^x

200

A particle  particle is moving right along the x axis and has a velocity modeled by the function v(t)=  x^2-4x+4 on the interval (0,4). When does the function decrease

(0,2)

300

A particle is moving right along the x axis and has a position function modeled by the equation y = 3cos(3x). Find all points in the interval (0,pi) where the function changes from increasing to decreasing or decreasing to increasing

pi/3 and 2pi/3

400

A particle is moving right along the x axis and has the position function s(t)= 3cos(x) +e^x+ 3tan(x). Find the acceleration function

-3cos(x)+e^x+6sec^2(x)tan(x)

500

A particle p is moving along the x axis. At time t=0, the particle's position s(t) is equal to 9. The velocity of the particle is given by the equation v(t)= 2x^2 +4. Write but do not solve an equation to find the position of the particle at t=5

s(0)+int(2x^2 +4) from 0 to 5