Primer on Limits and Derivatives
Basic definitions
The derivative of a function of one variable, f(x), is denoted
df
or f’(x), and is defined by the
dx
limit
f ( x) 
df
f ( x  x)  f ( x)
.
 lim
dx x0
x
On the graph of a function, the derivative is the slope of a line tangent to the graph at a given
value of x:
slope = f'(x)
y
y = f(x)
x
For a function of several variables, for example f(x,y,z), the partial derivatives with respect to
 f f
f
each variable are denoted
,
, and
, and they are defined by the limits
 x y
z
f
f ( x  x, y, z )  f ( x, y, z )
,
 lim
x x0
x
f
f ( x, y  y, z )  f ( x, y, z )
 lim
, and
y y 0
y
1
f
f ( x, y, z  z )  f ( x, y, z )
.
 lim
z z 0
z
Useful things to know about calculating limits
Limit of a sum or a difference:
If f ( x )  a and g ( x)  b as x  x0 , then f ( x)  g ( x)  a  b as x  x0 .
Limit of a product:
If f ( x )  a and g ( x)  b as x  x0 , then f ( x) g ( x)  ab as x  x0 .
Limit of a quotient:
If f ( x )  a and g ( x)  b as x  x0 , then
f ( x)
a
 as x  x0 .
g ( x)
b
L’Hospital’s rules are useful when the limit of a quotient involves indeterminate forms such as
0

or .
0

L’Hospital’s rule for
0
:
0
f ( x)
f ( x)
as x  x0 , provided
 lim
x  x0 g ( x )
x  x0 g ( x )
that the denominator g’(x) is not zero, and the ratio of derivatives goes to a definite limit.
If f ( x)  0 and g ( x)  0 as x  x0 , then lim
L’Hospital’s rule for

:

f ( x)
f ( x)
as x  x0 , provided
 lim
x  x0 g ( x )
x  x0 g ( x )
that the denominator g’(x) is not zero, and the ratio of derivatives goes to a definite limit.
If f ( x)   and g ( x)   as x  x0 , then lim
When taking limits as a variable x → ∞, it can be helpful to divide each term in an expression by
by the highest power of x, as in the two examples that follow.
Examples of limits
2
lim
R 
max R
KR
 lim
R 
max
K / R 1

max
1
 max
pmax I
pmax / I
 lim
0
2
2
I  k  I  I / K
I  k / I  1/ I  1/ K
I
I
lim
Calculating derivatives
Rather than calculating the limits involved in the definition of a derivative whenever a derivative
is needed, it is useful to have a list of derivatives of commonly used functions. Lists are found in
printed or electronic references such as the Handbook of Mathematical Functions, or the
Handbook of Chemistry and Physics. Mathematica, Wolfram Alpha, or other symbolic
computation programs can also be used to calculate derivatives.
Here are some of the rules for calculating ordinary derivatives of functions of one variable, f(x).
The same algebraic rules apply to partial derivatives ∂f / ∂x for functions of several variables,
treating the other variables (y, z, …) as algebraic constants.
Function
Derivative
f ( x )  g ( x )  h( x )
df dg dh


dx dx dx
f ( x )  g ( x ) h( x )
df dg
dh

h( x )  g ( x )
dx dx
dx
h( x )
g ( x)
f ( x) 
h( x )
f ( x) 
f ( x)  g  h( x) 
df dg dh

dx dh dx
f ( x)  const
df
0
dx
f ( x)  ax
df
 ax
dx
f ( x)  ag ( x)
df
dg
a
dx
dx
3
dg
dh
 g ( x)
dx
dx
2
 h( x ) 
f ( x)  ax n
df
 ax n 1
dx
f ( x)  x
df
1

dx 2 x
f ( x) 
1
x
df
1
 2
dx
x
f ( x) 
1
xn
df
n
  n 1
dx
x
f ( x) 
ax
b x
df
ab

dx (b  x) 2
f ( x) 
ax 2
b2  x2
df
2ab2 x

dx (b2  x 2 )2
f ( x)  ln x
df 1

dx x
f ( x)  ln g ( x)
df
1 dg

dx g ( x) dx
f ( x)  e x
df
 ex
dx
f ( x)  e g ( x )
df
dg
 ex
dx
dx
f ( x)  sin( x)
df
 cos( x)
dx
f ( x)  sin( g ( x))
df
dg
 cos( x)
dx
dx
4
f ( x)  cos( x)
df
  sin( x)
dx
f ( x)  cos( g ( x))
df
dg
  sin( x)
dx
dx
Formulas for derivatives and integrals
d
f ( x)dx  f ( x)
dx 
df
 dx dx  f ( x)  const
d q
f ( x)dx  f (q), for p constant and q variable
dq p
d q
f ( x)dx   f ( p), for p variable and q constant
dp p
Examples of derivatives:
1. Find the derivative of
f ( R) 
 max R
KR
 K
df
d  max R  ( K  R) max  max R max K  max R  max R


 max 2


2
2
dR dR  K  R 
( K  R)
( K  R)
(K  R)
2. Find the derivative of
f (I ) 
pmax I
k  I  I 2 / KI
pmax I
d 

dI  k  I  I 2 / K I

2
  k  I  I / K I  pmax  pmax I 1  2 I / K I 

2

k  I  I 2 / KI 
pmax k  pmax I  pmax I 2 / K I  pmax I  2 pmax I 2 / K I
k  I  I
2
/ KI 
2
5

pmax k  pmax I 2 / K I
k  I  I
2
/ KI 
2
3. Find the derivative of
f (t )  N 0 e rt
df
d
 N 0 ert (rt )  rN 0e rt
dt
dt
Exercises
1. Find the limit
lim
N 
 wN
1   w N
2. Find the limit
aN 2
N  b 2  N 2
lim
3. Find the limit
n0 e rt
t 
n0 n0 e rt
1 
K
K
lim
4. Find the derivative df/dN for
f (N ) 
 wN
1   w N
5. Find the derivative df/dN for
f (N ) 
aN 2
b2  N 2
6. Find the derivative df/dN for
 N
f ( N )  rN 1  
 K
6