Limits
lim(x->0)x^2-2
-2
x^2
2x
x^3
1/3x^2+c
cos(x)
-sin(x)
Chain Rule
f*g=(f'*g)g'
lim(x->0)(-(x^2-3x)/x)
3
x^3+x^2+x
3x^2+2x+1
ln(x^2)
1/(1/3x^3)+c
csc(x)
-csc(x)cot(x)
Product Rule
fg=g'f-f'g
lim(x->0+)(x^3-4)/4x
-infinity
sin(x)+cos(x)
e^(12x)
12xe^12x+C
cot(x)
-csc^2(x)
FTC
integral from a to b of f(x) = F(b)-F(a)
lim(x->Inf)sqrt(x)
1
f(x)/g(x)
(f(x)g'(x)-g(x)f'(x))/g(x)^2
(ln(x)/12)x
((x^2(2ln(x)-1))/48)+C
cot(x)csc(x)
-csc(x)(csc^2(x)+cot^2(x))
Intermediate Value Theorem
A function that is continuous between two points takes on every value between those two points
lim(x->0)(-6(x^3-6)/(4x^2+4x)
undefined
1/f(x)
-f'(x)/f(x)^2
x^2-e^(sqrt(5x))
(x^3/3)−2(5sqrt(x)−sqrt(5)esqrt(5)sqrt(x)/5^3/2+C
(cot(x)csc(x))/tan(x)
-csc(x)((csc^2(x)+cot^2(x))tan(x)+cot(x)sec^2(x))/tan^2(x)
Rolle's Theorem
When f(x) is both continuous and differentiable between a and b where f(a)=f(b) than at some point between a and b the derivative must be 0