Rules of Differentiation

Rules of Differentiation
Constant Rule
d c =0
dx ( )
Power Rule
d  xn  = nxn − 1
dx  
Constant Times a Function Rule
d cf = c df
dx
dx
Sum and Difference Rules
d (f ± g) = df ± dg
dx
dx dx
Product Rule
d (fg) = f dg + g df
dx
dx
dx
© Copyright 2010 by John Fetcho. All rights reserved.
Rules of Differentiation
Quotient Rule
df
dg
−
f
d  f  = dx dx
dx  g 
g2
g
Chain Rule
If y = f(u) and u = g(x), then
dy dy du
= •
dx du dx
General Power Rule
d  un  = nun−1 • du
dx  
dx
© Copyright 2010 by John Fetcho. All rights reserved.
Rules of Differentiation
Trigonometric Functions
d  sin u  = cos u • du

dx 
dx
d csc u = −csc u cot u • du
)
dx (
dx
d cos u = −sin u • du
)
dx (
dx
d sec u = sec u tan u • du
)
dx (
dx
d  tan u  = sec2 u • du

dx 
dx
d  cot u  = −csc2 u • du

dx 
dx
Exponential Functions
d  eu  = eu • du
dx  
dx
d  au  = au ln a • du
dx  
dx
Logarithmic Functions
d (ln u) = 1 • du
dx
u dx
d log u = 1 • du
a  u ln a dx
dx 
© Copyright 2010 by John Fetcho. All rights reserved.
Rules of Differentiation
Inverse Trigonometric Functions
d  sin−1 u  = 1 • du


dx 
1−u2 dx
d  csc −1 u  = − 1 • du


dx 
u u2 −1 dx
d  cos−1 u  = − 1 • du


dx 
1−u2 dx
d  sec −1 u  = 1 • du


dx 
u u2 −1 dx
d  tan−1 u  = 1 • du

 1+ u2 dx
dx 
d  cot −1 u  = − 1 • du


dx 
1+ u2 dx
Hyperbolic Functions
d  sinh u  = cosh u • du
( ) dx

dx 
d  csch u  = −csch u coth u • du
( )
( ) dx

dx 
d  cosh u  = sinh u • du
( ) dx

dx 
d  sech u  = −sech u tanh u • du
( )
( ) dx

dx 
d  tanh u  = sech2 u • du d  coth u  = −csch2 u • du
( ) dx dx 
( ) dx


dx 
© Copyright 2010 by John Fetcho. All rights reserved.