Calc

Calculus Fact Sheet

Essential Derivative Rules

d

dx(xn) = nxn−1 d

dx(ln(x)) = 1

x

d

dx(ex) = ex d

dx(bx) = bx ln(b)

d

dx(sin(x)) = cos(x) d

dx(tan(x)) = sec2(x) d

dx(sec(x)) = sec(x) tan(x)

d

dx(cos(x)) =−sin(x) d

dx(cot(x)) =−csc2(x) d

dx(csc(x)) =−csc(x) cot(x)

d

dx tan−1(x) =

1

x2 + 1

d

dx sin−1(x) =

1

√1−x2

d

dx sec−1(x) =

1

x√x2

−1

d

dx cot−1(x) =−1

x2 + 1

d

dx cos−1(x) =−1

√1−x2

d

dx csc−1(x) =−1

x√x2

−1

(FS)′

= FS′+ F′S N

′

D

=

DN′

−ND′

D2 [f(g(x))]′

= f′(g(x))g′(x)

Essential Integral Rules

1

xn dx=

n+ 1

xn+1 + C 1

1

ax+ bdx=

a

ln |ax+ b|+ C

1

eax dx=

a

eax + C bx dx=

1

ln(b) bx + C

1

cos(ax) dx=

asin(ax) + C sin(ax) dx=−

1

acos(ax) + C

sec2(x) dx= tan(x) + C csc2(x) dx=−cot(x) + C

sec(x) tan(x) dx= sec(x) + C csc(x) cot(x) dx=−csc(x) + C

1

1

x2 + a2 dx=

a

tan−1 x

a

+ C 1

√a2

−x2

dx= sin−1 x

a

+ C

tan(x) dx= ln |sec(x)|+ C cot(x) dx= ln |sin(x)|+ C

sec(x) dx= ln |sec(x) + tan(x)|+ C csc(x) dx= ln |csc(x)−cot(x)|+ C

1

sec3(x) dx=

2 sec(x) tan(x) + 1

ln |sec(x) + tan(x)|+ C

2

Precalculus Facts

π/6 π/4 x 0 π/3 π/2

sin(x) 0 1/2 √2/2 √3/2 1

sin2(x) + cos2(x) = 1 cos(x) 1 √3/2 √2/2 1/2 0

tan2(x) + 1 = sec2(x) 1 + cot2(x) = csc2(x)

sin2(x) = 1

2 (1−cos(2x)) cos2(x) = 1

2 (1 + cos(2x)) sin(x) cos(x) = 1

2 sin(2x)

ln(1) = 0 ln(e) = 1 ln(ab) = bln(a) ln(ab) = ln(a) + ln(b)

xaxb = xa+b (xa)b = xab n √x= x1/n 1

xa

= x−1