Calculus Cheat Sheet

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  • Topic: Derivative, Calculus, Mathematical analysis
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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
equivalent notations for the derivative.
df dy d
f ¢ ( x ) = y¢ =
=
= ( f ( x ) ) = Df ( x )
dx dx dx

If y = f ( x ) then,

If y = f ( x ) all of the following are equivalent
notations for derivative evaluated at x = a .
df
dy
f ¢ ( a ) = y ¢ x =a =
=
= Df ( a )
dx x =a dx x =a

Interpretation of the Derivative
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
time x then f ¢ ( a ) is the velocity of

equation of the tangent line at x = a is
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, 1.

( c f )¢ = c f ¢ ( x )

2.

( f ± g )¢ = f ¢ ( x ) ± g ¢ ( x )

3.

( f g )¢ =

æf
4. ç
èg

d
(c) = 0
dx
dn
6.
( x ) = n xn-1 – Power Rule
dx
d
7.
f ( g ( x )) = f ¢ ( g ( x )) g¢ ( x )
dx
This is the Chain Rule
5.

f ¢ g + f g ¢ – Product Rule

ö¢ f ¢ g - f g ¢
– Quotient Rule
÷=
g2
ø

(

)

Common Derivatives
d
( x) = 1
dx
d
( sin x ) = cos x
dx
d
( cos x ) = - sin x
dx
d
( tan x ) = sec2 x
dx
d
( sec x ) = sec x tan x
dx

d
( csc x ) = - csc x cot x
dx
d
( cot x ) = - csc2 x
dx
d
( sin -1 x ) = 1 2
dx
1- x
d
( cos-1 x ) = - 1 2
dx
1- x
d
1
( tan -1 x ) = 1 + x2
dx

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

dx
( a ) = a x ln ( a )
dx
dx
(e ) = ex
dx
d
1
( ln ( x ) ) = x , x > 0
dx
d
( ln x ) = 1 , x ¹ 0
dx
x
d
1
( log a ( x ) ) = x ln a , x > 0
dx

© 2005 Paul Dawkins

Calculus Cheat Sheet

Chain Rule Variants
The chain rule applied to some specific functions.
n
n -1
d
d
1.
é f ( x )ù = n é f ( x )ù f ¢ ( x )
5.
cos é f ( x ) ù = - f ¢ ( x ) sin é f ( x ) ù
ë
û
ë
û
ë
û
ë
û
dx
dx
d f ( x)
d
fx
e
tan é f ( x ) ù = f ¢ ( x ) sec 2 é f ( x ) ù
2.
= f ¢( x)e ( )
6.
ë
û
ë
û
dx
dx
d
f ¢( x)
d
7.
( sec [ f ( x)]) = f ¢( x) sec [ f ( x)] tan [ f ( x)]
3.
ln é f ( x ) ù =
ë
û
dx
dx
f ( x)
f ¢( x)
d
d
tan -1 é f ( x ) ù =
8.
2
4.
sin é f ( x ) ù = f ¢ ( x ) cos é f ( x ) ù
ë
û
ë
û
ë
û
dx
1 + é f ( x )ù
dx
ë
û

)

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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
¢ if e 2 x -9 y + x3 y 2 = sin ( y ) + 11x . Remember y = y ( x ) here, so products/quotients of x and y Find y
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 x3 y y¢ = cos ( y ) y¢ + 11 2e

2 x -9 y

- 9 y¢e

( 2 x y - 9e x
3

2 x -9 y

2 -9 y

+ 3x y + 2 x y y¢ = cos ( y ) y¢ + 11
2

2

3

- cos ( y ) ) y¢ = 11 - 2e2 x -9 y - 3x 2 y 2

Þ

11 - 2e 2 x -9 y - 3x 2 y 2
y¢ = 3
2 x y - 9e2 x -9 y - cos ( y )

Increasing/Decreasing – Concave Up/Concave Down
Critical Points
x = c is a critical point of f ( x ) provided either
1. f ¢ ( c ) = 0 or 2. f ¢ ( c ) doesn’t exist.
Increasing/Decreasing
1. If f ¢ ( x ) > 0 for all x in an interval I then...
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