Calculus Cheat Sheet
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
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