Calculus Cheat Sheet

Derivatives
Definition and Notation
f  x  h  f  x
If y  f  x  then the derivative is defined to be f   x   lim .
h0 h

If y  f  x  then all of the following are If y  f  x  all of the following are equivalent
equivalent notations for the derivative. notations for derivative evaluated at x  a .
df dy d df dy
f   x   y     f  x    Df  x  f   a   y x  a    Df  a 
dx dx dx dx x  a dx x  a

Interpretation of the Derivative

If y  f  x  then, 2. f   a  is the instantaneous rate of
1. m  f   a  is the slope of the tangent change of f  x  at x  a .
line to y  f  x  at x  a and the 3. If f  x  is the position of an object at
equation of the tangent line at x  a is time x then f   a  is the velocity of
given by y  f  a   f   a  x  a  . the object at x  a .

Basic Properties and Formulas

If f  x  and g  x  are differentiable functions (the derivative exists), c and n are any real numbers,
d
1.  c f   c f   x  5. c  0
dx
2. f  g   f   x   g   x  d n
6.
dx
 x   n x n 1 – Power Rule
3.  f g   f  g  f g  – Product Rule d
7.  
f  g  x   f   g  x  g  x 
 f  f  g  f g  dx
4.    – Quotient Rule This is the Chain Rule
g g2

Common Derivatives
d d d x
dx
 x  1
dx
 csc x    csc x cot x
dx
 a   a x ln  a 
d d d x
dx
 sin x   cos x
dx
 cot x    csc2 x
dx
 e   ex
d d 1 d 1
dx
 cos x    sin x
dx
 sin 1 x  
dx
 ln  x    , x  0
x
1  x2
d d 1
dx
 tan x   sec 2 x d
 cos 1 x   
1
dx
 ln x   x , x  0
dx 1  x2
d d 1
dx
 sec x   sec x tan x d
 tan 1 x  
1
dx
 log a  x   
x ln a
, x0
dx 1  x2

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 Calculus Cheat Sheet

Chain Rule Variants

The chain rule applied to some specific functions.
1.
d
dx
 n

 f  x    n  f  x  f   x 
n 1
5.
d
dx
 
cos  f  x     f   x  sin  f  x  

2.
dx
e 
d f  x

 f   x  e f  x 6.
d
dx
 
tan  f  x    f   x  sec 2  f  x  

f  x
 d
3.
d

ln  f  x     7.  sec  f ( x )   f ( x ) sec  f ( x)  tan  f ( x )
dx f  x dx
d f  x
4.
d
 
sin  f  x    f   x  cos  f  x   8. 
tan 1  f  x    
1   f  x 
2
dx dx

Higher Order Derivatives

The Second Derivative is denoted as The nth Derivative is denoted as
d2 f dn f
f   x   f  2   x   2 and is defined as f  n   x   n and is defined as
dx dx
f   x    f   x   , i.e. the derivative of the 
 
f  n   x   f  n 1  x  , i.e. the derivative of
first derivative, f   x  . the (n-1)st derivative, f  n 1 x .  
Implicit Differentiation

Find y if e 2 x 9 y
 x y  sin  y   11x . Remember y  y  x  here, so products/quotients of x and y
3 2

will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to
differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).
After differentiating solve for y .

e 2 x  9 y  2  9 y   3 x 2 y 2  2 x 3 y y  cos  y  y   11
11  2e 2 x  9 y  3x 2 y 2
2e 2 x9 y
 9 ye 2 x 9 y
 3 x y  2 x y y  cos  y  y   11
2 2 3
 y  3
2 x y  9e 2 x  9 y  cos  y 
 2 x y  9e x
3 2 9 y
 cos  y   y   11  2e 2 x  9 y  3 x 2 y 2

Increasing/Decreasing – Concave Up/Concave Down

Critical Points
x  c is a critical point of f  x  provided either Concave Up/Concave Down
1. If f   x   0 for all x in an interval I then
1. f   c   0 or 2. f   c  doesn’t exist.
f  x  is concave up on the interval I.
Increasing/Decreasing 2. If f   x   0 for all x in an interval I then
1. If f   x   0 for all x in an interval I then
f  x  is concave down on the interval I.
f  x  is increasing on the interval I.
2. If f   x   0 for all x in an interval I then Inflection Points
x  c is a inflection point of f  x  if the
f  x  is decreasing on the interval I.
concavity changes at x  c .
3. If f   x   0 for all x in an interval I then
f  x  is constant on the interval I.

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 Calculus Cheat Sheet

Extrema
Absolute Extrema Relative (local) Extrema
1. x  c is an absolute maximum of f  x  1. x  c is a relative (or local) maximum of
if f  c   f  x  for all x in the domain. f  x  if f  c   f  x  for all x near c.
2. x  c is a relative (or local) minimum of
2. x  c is an absolute minimum of f  x 
f  x  if f  c   f  x  for all x near c.
if f  c   f  x  for all x in the domain.
1st Derivative Test
Fermat’s Theorem If x  c is a critical point of f  x  then x  c is
If f  x  has a relative (or local) extrema at
1. a rel. max. of f  x  if f   x   0 to the left
x  c , then x  c is a critical point of f  x  .
of x  c and f   x   0 to the right of x  c .
Extreme Value Theorem 2. a rel. min. of f  x  if f   x   0 to the left
If f  x  is continuous on the closed interval of x  c and f   x   0 to the right of x  c .
 a, b then there exist numbers c and d so that, 3. not a relative extrema of f  x  if f   x  is
1. a  c, d  b , 2. f  c  is the abs. max. in the same sign on both sides of x  c .
 a, b , 3. f  d  is the abs. min. in  a, b . 2nd Derivative Test
If x  c is a critical point of f  x  such that
Finding Absolute Extrema
To find the absolute extrema of the continuous f   c   0 then x  c
function f  x  on the interval  a, b use the 1. is a relative maximum of f  x  if f   c   0 .
following process. 2. is a relative minimum of f  x  if f   c   0 .
1. Find all critical points of f  x  in  a, b . 3. may be a relative maximum, relative
2. Evaluate f  x  at all points found in Step 1. minimum, or neither if f   c   0 .
3. Evaluate f  a  and f  b  .
4. Identify the abs. max. (largest function Finding Relative Extrema and/or
value) and the abs. min.(smallest function Classify Critical Points
value) from the evaluations in Steps 2 & 3. 1. Find all critical points of f  x  .
2. Use the 1st derivative test or the 2nd
derivative test on each critical point.

Mean Value Theorem

If f  x  is continuous on the closed interval  a, b and differentiable on the open interval  a, b 
f b   f  a 
then there is a number a  c  b such that f   c   .
ba

Newton’s Method
f  xn 
If xn is the nth guess for the root/solution of f  x   0 then (n+1)st guess is xn 1  xn 
f   xn 
provided f   xn  exists.

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins