MFE Main Rules

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Last updated 10:10 PM on 4/8/26
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54 Terms

1
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ln(ab)
ln a + ln b
2
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ln(a/b)
ln a
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ln(a^r)
r ln a
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e^(ln x)
x (for x > 0)
5
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ln(e^x)
x
6
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ln(1)
0
7
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ln(e)
1
8
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ln(x) domain
x > 0
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ln(x) behavior
increasing function
10
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a^x
e^(x ln a)
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e^(a + b)
e^a * e^b
12
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e^(a
b)
13
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e^0
1
14
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e^(ln x)
x
15
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e^(
ln x)
16
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e^x range
always positive
17
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log equation solving
exponentiate both sides
18
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ln(expression) domain
expression > 0
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sqrt(expression) domain
expression >= 0
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denominator restriction
cannot equal 0
21
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sign analysis method
factor expression then test intervals
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positive times positive
positive
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negative times negative
positive
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positive times negative
negative
25
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critical points
where factors equal 0
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d/dx (x^n)
n x^(n
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d/dx (ln x)
1/x
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d/dx (e^x)
e^x
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d/dx (a^x)
a^x ln a
30
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product rule
(fg)' = f'g + fg'
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quotient rule
(f/g)' = (f'g
32
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chain rule
derivative of outside times derivative of inside
33
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d/dx e^(g(x))
e^(g(x)) * g'(x)
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log differentiation setup
ln y = g(x) ln f(x)
35
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inverse function identity
f(f^
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inverse derivative
(f^
37
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limit strategy
plug in first then simplify if needed
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0/0 indeterminate
factor or rewrite
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substitution trick
let u = e^x
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growth rates
ln x grows slowest, x^n middle, e^x fastest
41
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(ln x)^k / x^n as x→∞
0
42
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e^x minus polynomial as x→∞
infinity
43
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limit (x→0) (e^x
1)/x
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limit (x→0) (ln(1 + x))/x
1
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limit (x→0+) sqrt(x) ln x
0
46
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squeeze theorem
if a(x)
47
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exponential rewrite trick
rewrite powers using e and ln
48
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(1/x)^(ln x)
e^(
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e^(a ln b)
b^a
50
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tangent line formula
y
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tangent slope
derivative at point
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implicit differentiation
differentiate both sides and solve for y'
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ln(1/x)
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e^(a ln b)
b^a