AP Calculus AB Derivative and Integral review

Derivatives

100

f(x)=3x²+2x+1

f′(x)=6x+2

200

g(x)=5x²−4x+7

g′(x)=10x−4  

300

h(x)=x³+4x²−2x+5

h′(x)=3x²+8x−2

400

y(x)=e^2x

y′(x)=2e^2x

500

Given f(x)=x² and g(x)=e^x, find the derivative of g(f(x))

e^{x^2}⋅2x

Integrals

100

∫(3x²+2x+1)dx

x³+x²+x+C

200

∫(5x²−4x+7)dx

(5/3)x³−2x²+7x+C

300

∫(x³+4x²−2x+5)dx

(1/4)x⁴+(4/3)x³−x²+5x+C

400

∫e^xdx

e^x+C

500

∫1/(x²+1)dx

arctan(x)+C

Derivatives of Trig Functions

100

f(x)=sin(x)

f′(x)=cos⁡(x)

200

g(x)=3cos(x)

g′(x)=−3sin⁡(x)

300

h(x)=sin(x)cos(x)

h′(x)=(cos(x))²−(sin(x))²

400

y(x)=sin(x)/(cosx)

y'(x)=sec²(x)

500

z(x)=arctan⁡(x)

z′(x)=1/(1+x²)

Integrals of Trig Functions

100

∫sin(x)dx

−cos(x)+C

200

∫cos(x)dx

sin(x)+C

300

∫xsin(x)dx

−xcos(x)+sin(x)+C

400

∫sin²(x)dx

(1/2)x-(1/4)sin(2x)+C

500

∫1/(sqrt(1-x²))

arcsin(x)+C