Write:
Product rule:
Quotient rule:
Chain rule:
“Mini-chain” differentiation: what do these differentiate to for a constant c?
e^(c*x):
ln(c*x):
sin(c*x):
cos(c*x):
Mathematically describe:
1.
2.
3.
4.
5.
6.
7.
8.
f(x) is increasing:
f(x) is decreasing:
f’(x) is increasing:
f’(x) is decreasing:
Concave up:
Concave down:
Critical number:
Point of inflection:
Properties of:
Exponential function:
Log function:
Growth speed order of log, polynomial, and exponential:
Phrase
f(x)
f’(x)
f’’(x)
f(x) increasing
Graph upward
f’(x)>0
Graph above x-axis
f(x) decreasing
Graph downward
f’(x)<0
Graph below x-axis
f’(x) increasing
Graph concave up
Graph upward
f’’(x)>0
Graph above x-axis
f’(x) decreasing
Graph concave down
Graph downward
f’’(x)<0
Graph below x-axis
Critical points
“Flat” or undefined
slope of tangent
f’(x)= 0 or undefined
Local maximum
Graph Inc -> dec
f’(x)=0 and f’(x) + -> -
f’(x)=0 AND f’’(x)<0
Local minimum
Graph dec -> inc
f’(x)=0 and f’(x) - -> +
f’(x)=0 AND f’’(x)>0
Concave up
Graph opening
upward
Graph upward
f’’(x)>0
Graph below x-axis
Concave down
Graph opening
downward
Graph downward
f’’(x)<0
Graph below x-axis
Inflection point
Concavity changes
Looks like local
min/max
f’’(x)=0 (or und) AND
f’’(x) changes sign