AP Calculus AB Rules for Derivatives'
The derivative of a constant is zero.
d
-c=o
dx
d
-x
dx
n
n-i
=nx
The power rule, for any real number n
d
du
-(cu)=c-
Multiply the constant by the derivative of the
function.
~ (u ± v) = du ± dv
dx
dx dx
Differentiate a function term by term.
d
dv
du
-(uv) = u- + vdx
dx
dx
First times the derivative of the second plus the
second times the derivative of the first.
du
dv
v--u~(!!:)= dx 2 dx
dx V
v
(Denominator times the derivative of the
numerator minus the numerator times the
derivative of the denominator) +- denominator'
d
-sinx=cosx
dx
The derivative of sine is cosine.
dx
dx
I
The derivative of cosine is the opposite of sine.
d
-cosx
dx
= -sin x
d
-tanx
dx
= sec' x
The derivative of tangent is the square of
secant.
d
-cotx
dx
= =csc ' x
The derivative of cot~ngent is the opposite of
cosecant squared.
d
-secx
dx
= see x tan x
The derivative of secant is the product of
secant and tangent.
d
-cscx
dx
= -cscxcotx
The derivative of cosecant is the opposite of
the product of cosecant and cotangent.
-
dy = f'(g(x)). g'(x)
dx
d .
-SIn
dx
-1
d
-cos
dx
d
-tan
dx
d
-cot
dx
-1
-1
-1
x=~
x=
1
l-x2
-1
~1-x2
if Y = f(g(x))
Ixl< 1
Ixl< 1
Differentiate the outside functionf, evaluate it
at the inside function g, and multiply by the
derivative of the inside function. The chain
rule applies to all other derivative rules.
The derivative of inverse sine
The derivative of inverse cosine is the opposite
of the derivative of inverse sine.
The derivative of inverse tangent
1
x=-1+ X2
I
The derivative of invlrse cotangent is the
opposite of the derivJtive of inverse tangent.
-1
x=-1+X2
I
d
-sec
dx
d
-csc
dx
-1
1
x=
-1
-1
x=
Ixl> 1
Ixl~x2 -1
=ex
~bx
dx
= b" lnb
d
-log
dx
The derivative of invLse secant
Ixl~x2 -1
d
_ex
dx
d
-lnx=dx
Ixl> 1
1
x
1
x=-a
xlna
The derivative of cosecant is the opposite of the
derivative of secant.
The derivative of e' is itself.
b>O,b:td
x>o
a>O,a:;t:l
Note: u and v are differentiable functions ofx.
The derivative of an exponential function
whose base is other than e
The derivative of the natural logarithm function
The derivative of a logarithmic function whose
base is other than e