Derivative Rules

Derivative Rules

Derivative of a Constant Function:

The Power Rule:

d

(c) = 0, where c is a real number

dx

d n

(x ) = nxn−1 , where n is any real number

dx

The Constant Multiple Rule:

d

d

[cf(x)] = c f(x)

dx

dx

d

d

d

[f(x) + g(x)] =

f(x) + g(x)

dx

dx

dx

d

d

d

The Difference Rule:

[f(x) − g(x)] =

f(x) − g(x)

dx

dx

dx

The Sum Rule:

Derivative of ex :

d x

(e ) = ex

dx

The Product Rule:

d

d

d

[f(x)g(x)] = f(x) [g(x)] + g(x) [f(x)]

dx

dx

dx

d

d

g(x) [f(x)] − f(x) [g(x)]

d f(x)

dx

dx

The Quotient Rule:

=

dx g(x)

[g(x)]2

Derivatives of Trigonometric Functions:

d

d

(sin x) = cos x

(csc x) = − csc x cot x

dx

dx

d

d

(cos x) = − sin x

(sec x) = sec x tan x

dx

dx

d

d

(tan x) = sec2 x

(cot x) = − csc2 x

dx

dx

The Chain Rule:

d

f(g(x)) = f (g(x)) g (x)

dx

The Power Rule Combined with the Chain Rule:

The Exponential Rule:

d

[g(x)]n = n[g(x)]n−1 g (x)

dx

d x

(a ) = ax ln a

dx

Derivatives of Inverse Trigonometric Functions:

d

1

d

1

(sin−1 x) = √

(csc−1 x) = − √ 2

dx

dx

1 − x2

x x −1

d

1

d

1

(cos−1 x) = − √

(sec−1 x) = √ 2

dx

dx

1 − x2

x x −1

d

1

d

1

(tan−1 x) =

(cot−1 x) = −

2

dx

1+x

dx

1 + x2

1

Derivatives of Logarithmic Functions:

d

1

d

1

(loga x) =

(ln x) =

dx

x ln a

dx

x

Derivatives of Hyperbolic

d

(sinh x) = cosh x

dx

d

(cosh x) = sinh x

dx

d

(tanh x) = sech 2x

dx

d

1

(ln |x|) =

dx

x

Functions:

d

(csch x) = −csch x coth x

dx

d

(sech x) = −sech x tanh x

dx

d

(coth x) = −csch 2 x

dx

Derivatives of Inverse Hyperbolic Functions:

1

1

d

d

(sinh−1 x) = √

(csch−1 x) = − √ 2

dx

dx

1 + x2

|x| x + 1

1

d

1

d

(cosh−1 x) = √ 2

(sech−1 x) = − √

dx

dx

x −1

x 1 − x2

d

1

d

1

(tanh−1 x) =

(coth−1 x) =

2

dx

1−x

dx

1 − x2

2