AP Calculus AB: Unit 2 - Derivatives

Logs/Exponentials/ Inverse Trig

100

d/dtt^2lnt

2tlnt+t

200

If  y=e^(1/x)/x^2 

Then  y'= 

y'=(-e^(1/x)-2xe^(1/x))/x^4

300

d/dt (4log_3t-ln(8t^2))

4/(tln(3))-2/t

400

d/dx3^(arctan(5x))=

3^(arctan(5x))(ln3)(5/(1+25x^2))

500

If  f(x)=(5x)^(3x^2-2x+8) 

Then  f'(x)= 

lny=(3x^2-2x+8)ln(5x)

f'(x)=[(6x-2)ln(5x)+(3x^2-2x+8)(1/x)](5x)^(3x^2-2x+8)

Trig

100

d/dx(tanx/(1+cosx))

No need to simplify as long as the answer is one fraction

Anything equivalent to

((secx)^2(1+cosx)-tanx(-sinx))/(1+cosx)^2

200

If  f(x)=x^2sin(πx) 

Then  f'(x)= 

2xsin(πx)+πx^2cos(πx)

300

d/dxcot(3x^2+5)

-6xcsc^2(3x^2+5)

400

d/dxcsc^2(sinx)=

-2(csc^2(sinx))(cot(sinx))(cosx)

500

If  f(x)=sin(tansqrt(1+x^3)) 

Then 

f'(x)=

f'(x)=cos(tansqrt(1+x^3))(sec^2sqrt(1+x^3))((3x^2)/(2sqrt(1+x^3))

Motion

100

f(x)=(x^2-1)x(x+5)

f'(x)=3x^2+10x-1

200

y=2(6-x^2)^5

y'=10(6-x^2)^4x(-2x)

300

x^2ln(x)

x(2lnx+1)

400

f(x)=(x^2-2)(x^-1+2)

f'(x)4x+2x^-2+1

500

(2x^4-2x^2+2)(5x^4+1)

8x^7-60x^5+48x^3-4x

Related Rates

100

If x = 2, y = 1, and dx/dt = -4, find dy/dt for:

6y^2+x^2=2-x^3e^(4-4y)

dy/dt=8/11

200

A rectangle's length is always 3 times its width.  If the shorter side is decreasing at a rate of 2 inches per minute, at what rate is the area decreasing? (include units)

The area is decreasing at a rate of 72 square inches per minute.

(Changing at -72 square inches per minute)

300

20x^2

40x

400

5x^3 + 10x^2 + 50x

15x^2 + 20x + 50

500

57x^7-23x^5-15x^3+8x+2

399x^6-115x^4-45x^2+8

Halloween Trivia

100

The name of the villain in "The Nightmare Before Christmas"

Oogie Boogie

200

another name for a "lycanthrope"

werewolf

300

The author of the 1818 novel, "Frankenstein"

Mary Shelley

400

Rotten Tomatoes lists this movie as the #1 scariest of all time

The Exorcist

500

The most popular Halloween costume in the US in 2023

Barbie