Functions – Derivatives and Antiderivatives Function Derivative Power Function: f x = x n f ʹ′ x = nx n−1 ()
()
()
()
()
n ≠ −1
n = −1
f ʹ′ x = lnb bx bx
F x =
lnb
f ʹ′ x = e x €
1
f ʹ′ x =
lnb x
F x = e x () ( )
()
()
€
()
( )
f ʹ′ x = ( )
€
1
x
€
f ʹ′ x = cos x ()
f ʹ′ x = −sin x ()
f ʹ′ x = sec2 x ()
f ʹ′ x = sec x tan x ()
€
f ʹ′ x = −csc x cot x €
f ʹ′(x ) = −csc2 x €
()
()
()
()
F x =
1
lnb
(x ln x − x ) F x = x ln x − x ()
F x = −cos x ()
€
F x = sin x ()
€
€
()
()
Note: A€
ll the antiderivatives would have €
“+ C” for completeness. ()
()
()
⎧⎪ 1 x n+1
F x = ⎨ n+1
⎪⎩ln x
()
Exponential Function: f = bx €
€
Natural Exponential: f x = e x €
€
Logarithm Function: f x = log b x €
€
Natural Logarithm: f x = ln x €
€
f x = sin x €
€
f x = cos x €
€
f x = tan x €
€
f x = sec x €
€
f x = csc x €
€
f x = cot x €
€
()
Antiderivative €
F x = −lncos x ()
F x = lnsec x + tan x ()
F x = −lncsc x + cot x ()
F x = lnsin x ()