**f( x )= y → f ′ ( x )=
dy/dx **

**f( x )= x ^{n} → f ′ ( x )= nx^{n-1}**

**f( x )=
constant → f ′ ( x )= 0 **

**f( x )= x →
f ′ ( x )= 1**

**f( x )= e ^{x} → f ′ ( x )= e^{x}**

**f( x )= e ^{ax} → f ′ ( x )= a e^{ax}**

**f( x )= logx
→ f ′ ( x )= 1/x**

**f( x )= sinx
→ f ′ ( x )= cosx**

**f( x )= cosx
→ f ′ ( x )= - sinx**

**f( x )= tanx
→ f ′ ( x )= sec ^{2} x**

**f( x )= secx
→ f ′ ( x )= secx tanx**

**f( x )=
cosecx → f ′ ( x )= -cosecx cotx **

**f( x )= cotx
→ f ′ ( x )= - cosec ^{2} x**

**f( x )=
sin ^{-1} x → f ′
( x )= 1/√(1- x^{2})**

**f( x )=
cos ^{-1} x → f ′
( x )= - 1/√(1+x^{2})**

**f( x )=
tan ^{-1} x → f ′
( x )= 1/(1+x^{2})**

**f( x )=
cot ^{-1} x → f ′
( x )= - 1/(1+x^{2})**

**f( x )=
cosec ^{-1} x → f
′ ( x )= - 1/|x| √(x^{2 }-1)**

**f( x )=
Sec ^{-1} x → f ′
( x )= 1/|x| √(x^{2 }-1)**

**f( x )= UV →
f ′ ( x )= UV′ + VU′(Product rule)**

**f( x )= U/V →
f ′ ( x )= (VU′ - UV′) / V ^{2} (Quotient rule)**

**f( x )= f(g(x))
→ f ′ ( x )= f ′ (g) * g′ (x) (Chain rule)**

**Partial Differentiation
Identities**

**if f( x(r,s),
y(r,s) )**

**df / dr = df / dx * dx / DR + df / dy *
dy / DR**

**df / ds = df / dx * dx / Ds + df / dy *
dy / Ds**

**if f( x(r,s) )**

**df / DR = df / dx * dx / DR**

**df / Ds = df / dx * dx / Ds**

**directional
derivative**

**df(x,y) / d(Xi sub a) = f1(x,y) cos(a)
+ f2(x,y) sin(a)**

**(Xi sub a) = angle counterclockwise
from pos. x axis.**

**Asymptotes **

**Definition
of a horizontal asymptote: The line y = y _{0} is a
"horizontal asymptote" of f(x) if and only if f(x)
approaches y_{0} as x approaches + or - ∞. **

**Definition
of a vertical asymptote: The line x = x _{0} is a
"vertical asymptote" of f(x) if and only if f(x) approaches
+ or - ∞as x approaches x_{0} from the left or from the right. **

**Definition
of a slant asymptote: the line y = ax + b is a "slant
asymptote" of f(x) if and only if lim _{(x-->+/-}∞_{)} f(x) = ax + b. **

**Concavity **

**Definition
of a concave up curve: f(x) is "concave up" at x _{0} if and only if f '(x) is increasing at x_{0} **

**Definition
of a concave down curve: f(x) is "concave down" at x _{0} if and only if f '(x) is decreasing at x_{0} **

**The
second derivative test: If f ''(x) exists at x _{0} and is
positive, then f ''(x) is concave up at x_{0}. If f ''(x_{0})
exists and is negative, then f(x) is concave down at x_{0}.
If f ''(x) does not exist or is zero, then the test fails. **

**Critical
Points **

**Definition
of a critical point: a critical point on f(x) occurs at x _{0} if and only if either f '(x_{0}) is zero or the derivative
doesn't exist. **

**Extrema
(Maxima and Minima) **

**Local
(Relative) Extrema **

**Definition
of a local maxima: A function f(x) has a local maximum at x _{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) >= f(x) for all x in I. **

**Definition
of a local minima: A function f(x) has a local minimum at x _{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) <= f(x) for all x in I. **

**Occurrence
of local extrema: All local extrema occur at critical points, but
not all critical points occur at local extrema. **

**The
first derivative test for local extrema: If f(x) is increasing (f
'(x) > 0) for all x in some interval (a, x _{0}] and f(x)
is decreasing (f '(x) < 0) for all x in some interval [x_{0},
b), then f(x) has a local maximum at x_{0}. If f(x) is
decreasing (f '(x) < 0) for all x in some interval (a, x_{0}]
and f(x) is increasing (f '(x) > 0) for all x in some interval
[x_{0}, b), then f(x) has a local minimum at x_{0}. **

**The
second derivative test for local extrema: If f '(x _{0}) =
0 and f ''(x_{0}) > 0, then f(x) has a local minimum at
x_{0}. If f '(x_{0}) = 0 and f ''(x_{0}) <
0, then f(x) has a local maximum at x_{0}. **

**Absolute
Extrema **

**Definition
of absolute maxima: y _{0} is the "absolute maximum"
of f(x) on I if and only if y_{0} >= f(x) for all x on I. **

**Definition
of absolute minima: y _{0} is the "absolute minimum"
of f(x) on I if and only if y_{0} <= f(x) for all x on I. **

**The
extreme value theorem: If f(x) is continuous in a closed interval
I, then f(x) has at least one absolute maximum and one absolute
minimum in I. **

**Occurrence
of absolute maxima: If f(x) is continuous in a closed interval I,
then the absolute maximum of f(x) in I is the maximum value of f(x)
on all local maxima and endpoints on I. **

**Occurrence
of absolute minima: If f(x) is continuous in a closed interval I,
then the absolute minimum of f(x) in I is the minimum value of f(x)
on all local minima and endpoints on I. **

**Alternate
method of finding extrema: If f(x) is continuous in a closed
interval I, then the absolute extrema of f(x) in I occur at the
critical points and/or at the endpoints of I. (This is a less
specific form of the above.) **

**Increasing/Decreasing
Functions **

**Definition
of an increasing function: A function f(x) is "increasing"
at a point x _{0} if and only if there exists some interval I
containing x_{0} such that f(x_{0}) > f(x) for all
x in I to the left of x_{0} and f(x_{0}) < f(x)
for all x in I to the right of x_{0}. **

**Definition
of a decreasing function: A function f(x) is "decreasing"
at a point x _{0} if and only if there exists some interval I
containing x_{0} such that f(x_{0}) < f(x) for all
x in I to the left of x_{0} and f(x_{0}) > f(x)
for all x in I to the right of x_{0}. **

**The
first derivative test: If f '(x _{0}) exists and is
positive, then f '(x) is increasing at x_{0}. If f '(x)
exists and is negative, then f(x) is decreasing at x_{0}. If
f '(x_{0}) does not exist or is zero, then the test tells
fails. **

**Inflection
Points **

**Definition
of an inflection point: An inflection point occurs on f(x) at x _{0} if and only if f(x) has a tangent line at x_{0} and there
exists and interval I containing x_{0} such that f(x) is
concave up on one side of x_{0} and concave down on the other
side. **