Differentiation rules

Rules for computing derivatives of functions

Part of a series of articles about

Calculus
∫ a b f ′ ( t ) d t = f ( b ) − f ( a ) {\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

* Fundamental theorem

* Limits
* Continuity

* Rolle's theorem
* Mean value theorem
* Inverse function theorem

Differential

|  Definitions

* Derivative (generalizations)
* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Implicit differentiation
* Logarithmic differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* L'Hôpital's rule
* Inverse
* General Leibniz
* Faà di Bruno's formula
* Reynolds

Integral

* Lists of integrals
* Integral transform
* Leibniz integral rule

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration
* Integral of inverse functions

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric, tangent half-angle, Euler)
* Euler's formula
* Partial fractions (Heaviside's method)
* Changing order
* Reduction formulae
* Differentiating under the integral sign
* Risch algorithm

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Gradient
* Green's
* Stokes'
* Divergence
* Generalized Stokes
* Helmholtz decomposition

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Advanced

* Calculus on Euclidean space
* Generalized functions
* Limit of distributions

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Miscellanea

* Precalculus
* History
* Glossary
* List of topics
* Integration Bee
* Mathematical analysis
* Nonstandard analysis

* v
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* e

This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

## Elementary rules of differentiation

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Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb {R} } ) that return real values, although, more generally, the formulas below apply wherever they are well defined,[1][2] including the case of complex numbers ( C {\textstyle \mathbb {C} } ).[3]

### Constant term rule

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For any value of  c {\textstyle c} , where  c ∈ R {\textstyle c\in \mathbb {R} } , if  f ( x ) {\textstyle f(x)} is the constant function given by  f ( x ) = c {\textstyle f(x)=c} , then  d f d x = 0 {\textstyle {\frac {df}{dx}}=0} .[4]

#### Proof

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Let  c ∈ R {\textstyle c\in \mathbb {R} } and  f ( x ) = c {\textstyle f(x)=c} . By the definition of the derivative:  f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 ( c ) − ( c ) h = lim h → 0 0 h = lim h → 0 0 = 0. {\displaystyle {\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\\&=\lim _{h\to 0}{\frac {(c)-(c)}{h}}\\\&=\lim _{h\to 0}{\frac {0}{h}}\\\&=\lim _{h\to 0}0\\\&=0.\end{aligned}}}

This computation shows that the derivative of any constant function is 0.

#### Intuitive (geometric) explanation

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The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

In other words, the value of the constant function,  y {\textstyle y} , will not change as the value of  x {\textstyle x} increases or decreases.

At each point, the derivative is the slope of a line that is tangent to the curve at that point. Note: the derivative at point A is positive where green and dash–dot, negative where red and dashed, and 0 where black and solid.

### Differentiation is linear

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Main article: Linearity of differentiation

For any functions  f {\textstyle f} and  g {\textstyle g} and any real numbers  a {\textstyle a} and  b {\textstyle b} , the derivative of the function  h ( x ) = a f ( x ) + b g ( x ) {\textstyle h(x)=af(x)+bg(x)} with respect to  x {\textstyle x} is  h ′ ( x ) = a f ′ ( x ) + b g ′ ( x ) {\textstyle h'(x)=af'(x)+bg'(x)} .

In Leibniz's notation, this formula is written as:  d ( a f + b g ) d x = a d f d x + b d g d x . {\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}

Special cases include:

* The constant factor rule:

( a f ) ′ = a f ′ , {\displaystyle (af)'=af',}

* The sum rule:

( f + g ) ′ = f ′ + g ′ , {\displaystyle (f+g)'=f'+g',}

* The difference rule:

( f − g ) ′ = f ′ − g ′ . {\displaystyle (f-g)'=f'-g'.}

### Product rule

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Main article: Product rule

For the functions  f {\textstyle f} and  g {\textstyle g} , the derivative of the function  h ( x ) = f ( x ) g ( x ) {\textstyle h(x)=f(x)g(x)} with respect to  x {\textstyle x} is:  h ′ ( x ) = ( f g ) ′ ( x ) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) . {\displaystyle h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).}

In Leibniz's notation, this formula is written:  d ( f g ) d x = g d f d x + f d g d x . {\displaystyle {\frac {d(fg)}{dx}}=g{\frac {df}{dx}}+f{\frac {dg}{dx}}.}

### Chain rule

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Main article: Chain rule

The derivative of the function  h ( x ) = f ( g ( x ) ) {\textstyle h(x)=f(g(x))} is:  h ′ ( x ) = f ′ ( g ( x ) ) ⋅ g ′ ( x ) . {\displaystyle h'(x)=f'(g(x))\cdot g'(x).}

In Leibniz's notation, this formula is written as:  d d x h ( x ) = d d z f ( z ) | z = g ( x ) ⋅ d d x g ( x ) , {\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),} often abridged to:  d h ( x ) d x = d f ( g ( x ) ) d g ( x ) ⋅ d g ( x ) d x . {\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}

Focusing on the notion of maps, and the differential being a map  D {\textstyle {\text{D}}} , this formula is written in a more concise way as:  [ D ( f ∘ g ) ] x = [ D f ] g ( x ) ⋅ [ D g ] x . {\displaystyle [{\text{D}}(f\circ g)]_{x}=[{\text{D}}f]_{g(x)}\cdot [{\text{D}}g]_{x}.}

### Inverse function rule

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Main article: Inverse function rule

If the function  f {\textstyle f} has an inverse function g {\textstyle g} , meaning that  g ( f ( x ) ) = x {\textstyle g(f(x))=x} and  f ( g ( y ) ) = y {\textstyle f(g(y))=y} , then:  g ′ = 1 f ′ ∘ g . {\displaystyle g'={\frac {1}{f'\circ g}}.}

In Leibniz notation, this formula is written as:  d x d y = 1 d y d x . {\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.}

## Power laws, polynomials, quotients, and reciprocals

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### Polynomial or elementary power rule

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Main article: Power rule

If  f ( x ) = x r {\textstyle f(x)=x^{r}} , for any real number  r ≠ 0 {\textstyle r\neq 0} , then:  f ′ ( x ) = r x r − 1 . {\displaystyle f'(x)=rx^{r-1}.}

When  r = 1 {\textstyle r=1} , this formula becomes the special case that, if  f ( x ) = x {\textstyle f(x)=x} , then  f ′ ( x ) = 1 {\textstyle f'(x)=1} .

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

### Reciprocal rule

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Main article: Reciprocal rule

The derivative of  h ( x ) = 1 f ( x ) {\textstyle h(x)={\frac {1}{f(x)}}} for any (nonvanishing) function  f {\textstyle f} is:  h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 , {\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},} wherever  f {\textstyle f} is nonzero.

In Leibniz's notation, this formula is written:  d ( 1 f ) d x = − 1 f 2 d f d x . {\displaystyle {\frac {d\left({\frac {1}{f}}\right)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.}

The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule.

### Quotient rule

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Main article: Quotient rule

If  f {\textstyle f} and  g {\textstyle g} are functions, then:  ( f g ) ′ = f ′ g − g ′ f g 2 , {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}},} wherever  g {\textstyle g} is nonzero.

This can be derived from the product rule and the reciprocal rule.

### Generalized power rule

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Main article: Power rule

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions  f {\textstyle f} and  g {\textstyle g} ,  ( f g ) ′ = ( e g ln ⁡ f ) ′ = f g ( f ′ g f + g ′ ln ⁡ f ) , {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\quad } wherever both sides are well defined.

Special cases:

* If  f ( x ) = x a {\textstyle f(x)=x^{a}} , then  f ′ ( x ) = a x a − 1 {\textstyle f'(x)=ax^{a-1}} when  a {\textstyle a} is any nonzero real number and  x {\textstyle x} is positive.
* The reciprocal rule may be derived as the special case where  g ( x ) = − 1 {\textstyle g(x)=-1\\!} .

## Derivatives of exponential and logarithmic functions

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d d x ( c a x ) = a c a x ln ⁡ c , c > 0. {\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0.} The equation above is true for all  c {\displaystyle c} , but the derivative for  c < 0 {\displaystyle c<0} yields a complex number.

d d x ( e a x ) = a e a x . {\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}.}

d d x ( log c ⁡ x ) = 1 x ln ⁡ c , c > 1. {\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1.} The equation above is also true for all  c {\textstyle c} but yields a complex number if  c < 0 {\textstyle c<0} .

d d x ( ln ⁡ x ) = 1 x , x > 0. {\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.}

d d x ( ln ⁡ | x | ) = 1 x , x ≠ 0. {\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.}

d d x ( W ( x ) ) = 1 x + e W ( x ) , x > − 1 e , {\displaystyle {\frac {d}{dx}}\left(W(x)\right)={1 \over {x+e^{W(x)}}},\qquad x>-{1 \over e},} where  W ( x ) {\textstyle W(x)} is the Lambert W function.

d d x ( x x ) = x x ( 1 + ln ⁡ x ) . {\displaystyle {\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).}

d d x ( f ( x ) g ( x ) ) = g ( x ) f ( x ) g ( x ) − 1 d f d x + f ( x ) g ( x ) ln ⁡ ( f ( x ) ) d g d x , if  f ( x ) > 0  and  d f d x  and  d g d x  exist. {\displaystyle {\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0{\text{ and }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}}

d d x ( f 1 ( x ) f 2 ( x ) ( . . . ) f n ( x ) ) = [ ∑ k = 1 n ∂ ∂ x k ( f 1 ( x 1 ) f 2 ( x 2 ) ( . . . ) f n ( x n ) ) ] | x 1 = x 2 = . . . = x n = x ,  if  f i < n ( x ) > 0  and  d f i d x  exists. {\displaystyle {\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},\qquad {\text{ if }}f_{i<n}(x)>0{\text{ and }}{\frac {df_{i}}{dx}}{\text{ exists.}}}

### Logarithmic derivatives

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The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):  ( ln ⁡ f ) ′ = f ′ f , {\displaystyle (\ln f)'={\frac {f'}{f}},} wherever  f {\textstyle f} is positive.

Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.[citation needed]

Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified expression for taking derivatives.

## Derivatives of trigonometric functions

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Main article: Differentiation of trigonometric functions

d d x sin ⁡ x = cos ⁡ x {\displaystyle {\frac {d}{dx}}\sin x=\cos x} |  d d x arcsin ⁡ x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arcsin x={\frac {1}{\sqrt {1-x^{2}}}}}

|

d d x cos ⁡ x = − sin ⁡ x {\displaystyle {\frac {d}{dx}}\cos x=-\sin x} |  d d x arccos ⁡ x = − 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arccos x=-{\frac {1}{\sqrt {1-x^{2}}}}}
d d x tan ⁡ x = sec 2 ⁡ x = 1 cos 2 ⁡ x = 1 + tan 2 ⁡ x {\displaystyle {\frac {d}{dx}}\tan x=\sec ^{2}x={\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x} |  d d x arctan ⁡ x = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\arctan x={\frac {1}{1+x^{2}}}}
d d x csc ⁡ x = − csc ⁡ x cot ⁡ x {\displaystyle {\frac {d}{dx}}\csc x=-\csc {x}\cot {x}} |  d d x arccsc ⁡ x = − 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arccsc} x=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x sec ⁡ x = sec ⁡ x tan ⁡ x {\displaystyle {\frac {d}{dx}}\sec x=\sec {x}\tan {x}} |  d d x arcsec ⁡ x = 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x cot ⁡ x = − csc 2 ⁡ x = − 1 sin 2 ⁡ x = − 1 − cot 2 ⁡ x {\displaystyle {\frac {d}{dx}}\cot x=-\csc ^{2}x=-{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x} |  d d x arccot ⁡ x = − 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arccot} x=-{1 \over 1+x^{2}}}

The derivatives in the table above are for when the range of the inverse secant is  [ 0 , π ] {\textstyle [0,\pi ]} and when the range of the inverse cosecant is  [ − π 2 , π 2 ] {\textstyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]} .

It is common to additionally define an inverse tangent function with two arguments,  arctan ⁡ ( y , x ) {\textstyle \arctan(y,x)} . Its value lies in the range  [ − π , π ] {\textstyle [-\pi ,\pi ]} and reflects the quadrant of the point  ( x , y ) {\textstyle (x,y)} . For the first and fourth quadrant (i.e.,  x > 0 {\displaystyle x>0} ), one has  arctan ⁡ ( y , x > 0 ) = arctan ⁡ ( y x ) {\textstyle \arctan(y,x>0)=\arctan({\frac {y}{x}})} . Its partial derivatives are:  ∂ arctan ⁡ ( y , x ) ∂ y = x x 2 + y 2 and ∂ arctan ⁡ ( y , x ) ∂ x = − y x 2 + y 2 . {\displaystyle {\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}\qquad {\text{and}}\qquad {\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.}

## Derivatives of hyperbolic functions

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Main article: Hyperbolic functions § Derivatives

d d x sinh ⁡ x = cosh ⁡ x {\displaystyle {\frac {d}{dx}}\sinh x=\cosh x} |  d d x arsinh ⁡ x = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arsinh} x={\frac {1}{\sqrt {1+x^{2}}}}}

|

d d x cosh ⁡ x = sinh ⁡ x {\displaystyle {\frac {d}{dx}}\cosh x=\sinh x} |  d d x arcosh ⁡ x = 1 x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcosh} x={\frac {1}{\sqrt {x^{2}-1}}}}
d d x tanh ⁡ x = sech 2 ⁡ x = 1 − tanh 2 ⁡ x {\displaystyle {\frac {d}{dx}}\tanh x={\operatorname {sech} ^{2}x}=1-\tanh ^{2}x} |  d d x artanh ⁡ x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {artanh} x={\frac {1}{1-x^{2}}}}
d d x csch ⁡ x = − csch ⁡ x coth ⁡ x {\displaystyle {\frac {d}{dx}}\operatorname {csch} x=-\operatorname {csch} {x}\coth {x}} |  d d x arcsch ⁡ x = − 1 | x | 1 + x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arcsch} x=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}}
d d x sech ⁡ x = − sech ⁡ x tanh ⁡ x {\displaystyle {\frac {d}{dx}}\operatorname {sech} x=-\operatorname {sech} {x}\tanh {x}} |  d d x arsech ⁡ x = − 1 x 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arsech} x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}}
d d x coth ⁡ x = − csch 2 ⁡ x = 1 − coth 2 ⁡ x {\displaystyle {\frac {d}{dx}}\coth x=-\operatorname {csch} ^{2}x=1-\coth ^{2}x} |  d d x arcoth ⁡ x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arcoth} x={\frac {1}{1-x^{2}}}}

## Derivatives of special functions

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### Gamma function

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Main article: Gamma function

Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,dt} Γ ′ ( x ) = ∫ 0 ∞ t x − 1 e − t ln ⁡ t d t = Γ ( x ) ( ∑ n = 1 ∞ ( ln ⁡ ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) , {\displaystyle {\begin{aligned}\Gamma '(x)&=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt\\\&=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\\&=\Gamma (x)\psi (x),\end{aligned}}} with  ψ ( x ) {\textstyle \psi (x)} being the digamma function, expressed by the parenthesized expression to the right of  Γ ( x ) {\textstyle \Gamma (x)} in the line above.

### Riemann zeta function

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Main article: Riemann zeta function

ζ ( x ) = ∑ n = 1 ∞ 1 n x {\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}} ζ ′ ( x ) = − ∑ n = 1 ∞ ln ⁡ n n x = − ln ⁡ 2 2 x − ln ⁡ 3 3 x − ln ⁡ 4 4 x − ⋯ = − ∑ p  prime p − x ln ⁡ p ( 1 − p − x ) 2 ∏ q  prime , q ≠ p 1 1 − q − x {\displaystyle {\begin{aligned}\zeta '(x)&=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \\\&=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\end{aligned}}}

## Derivatives of integrals

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Main article: Differentiation under the integral sign

Suppose that it is required to differentiate with respect to  x {\textstyle x} the function:  F ( x ) = ∫ a ( x ) b ( x ) f ( x , t ) d t , {\displaystyle F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,}

where the functions  f ( x , t ) {\textstyle f(x,t)} and  ∂ ∂ x f ( x , t ) {\textstyle {\frac {\partial }{\partial x}}\,f(x,t)} are both continuous in both  t {\textstyle t} and  x {\textstyle x} in some region of the  ( t , x ) {\textstyle (t,x)} plane, including  a ( x ) ≤ t ≤ b ( x ) {\textstyle a(x)\leq t\leq b(x)} , where  x 0 ≤ x ≤ x 1 {\textstyle x_{0}\leq x\leq x_{1}} , and the functions  a ( x ) {\textstyle a(x)} and  b ( x ) {\textstyle b(x)} are both continuous and both have continuous derivatives for  x 0 ≤ x ≤ x 1 {\textstyle x_{0}\leq x\leq x_{1}} . Then, for  x 0 ≤ x ≤ x 1 {\textstyle \,x_{0}\leq x\leq x_{1}} :  F ′ ( x ) = f ( x , b ( x ) ) b ′ ( x ) − f ( x , a ( x ) ) a ′ ( x ) + ∫ a ( x ) b ( x ) ∂ ∂ x f ( x , t ) d t . {\displaystyle F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.}

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

## Derivatives to nth order

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Some rules exist for computing the  n {\textstyle n} th derivative of functions, where  n {\textstyle n} is a positive integer, including:

### Faà di Bruno's formula

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Main article: Faà di Bruno's formula

If  f {\textstyle f} and  g {\textstyle g} are  n {\textstyle n} -times differentiable, then:  d n d x n [ f ( g ( x ) ) ] = n ! ∑ { k m } f ( r ) ( g ( x ) ) ∏ m = 1 n 1 k m ! ( g ( m ) ( x ) ) k m , {\displaystyle {\frac {d^{n}}{dx^{n}}}[f(g(x))]=n!\sum _{\\{k_{m}\\}}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}},} where  r = ∑ m = 1 n − 1 k m {\textstyle r=\sum _{m=1}^{n-1}k_{m}} and the set  { k m } {\textstyle \\{k_{m}\\}} consists of all non-negative integer solutions of the Diophantine equation ∑ m = 1 n m k m = n {\textstyle \sum _{m=1}^{n}mk_{m}=n} .

### General Leibniz rule

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Main article: General Leibniz rule

If  f {\textstyle f} and  g {\textstyle g} are  n {\textstyle n} -times differentiable, then:  d n d x n [ f ( x ) g ( x ) ] = ∑ k = 0 n ( n k ) d n − k d x n − k f ( x ) d k d x k g ( x ) . {\displaystyle {\frac {d^{n}}{dx^{n}}}[f(x)g(x)]=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {d^{n-k}}{dx^{n-k}}}f(x){\frac {d^{k}}{dx^{k}}}g(x).}

## See also

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* Differentiable function – Mathematical function whose derivative exists
* Differential of a function – Notion in calculus
* Differentiation of integrals – Problem in mathematics
* Differentiation under the integral sign – Differentiation under the integral sign formulaPages displaying short descriptions of redirect targets
* Hyperbolic functions – Collective name of 6 mathematical functions
* Inverse hyperbolic functions – Mathematical functions
* Inverse trigonometric functions – Inverse functions of sin, cos, tan, etc.
* Lists of integrals
* List of mathematical functions
* Matrix calculus – Specialized notation for multivariable calculus
* Trigonometric functions – Functions of an angle
* Vector calculus identities – Mathematical identities

## References

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1. ^ Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, ISBN 978-0-07-150861-2.
2. ^ Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, ISBN 978-0-07-162366-7.
3. ^ Complex Variables, M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3
4. ^ "Differentiation Rules". University of Waterloo – CEMC Open Courseware. Retrieved 3 May 2022.

## Sources and further reading

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These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in:

* Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISBN 978-0-07-154855-7.
* The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
* Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
* NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, ISBN 978-0-521-19225-5.

## External links

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Library resources about
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* Resources in your library

* Derivative calculator with formula simplification
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* Related rates
* Stationary points
* First derivative test
* Second derivative test
* Extreme value theorem
* Maximum and minimum
* Further applications
* Newton's method
* Taylor's theorem
* Differential equation
* Ordinary differential equation
* Partial differential equation
* Stochastic differential equation

Integral calculus|

* Antiderivative
* Arc length
* Riemann integral
* Basic properties
* Constant of integration
* Fundamental theorem of calculus
* Differentiating under the integral sign
* Integration by parts
* Integration by substitution
* trigonometric
* Euler
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Trapezoidal rule
* Volumes
* Washer method
* Shell method
* Integral equation
* Integro-differential equation

Vector calculus|

* Derivatives
* Curl
* Directional derivative
* Divergence
* Gradient
* Laplacian
* Basic theorems
* Line integrals
* Green's
* Stokes'
* Gauss'

Multivariable calculus|

* Divergence theorem
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Lagrange multiplier
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Surface integral
* Volume integral
* Advanced topics
* Differential forms
* Exterior derivative
* Generalized Stokes' theorem
* Tensor calculus

Sequences and series|

* Arithmetico-geometric sequence
* Types of series
* Alternating
* Binomial
* Fourier
* Geometric
* Harmonic
* Infinite
* Power
* Maclaurin
* Taylor
* Telescoping
* Tests of convergence
* Abel's
* Alternating series
* Cauchy condensation
* Direct comparison
* Dirichlet's
* Integral
* Limit comparison
* Ratio
* Root
* Term

Special functions
and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

| Integrals|

* rational functions
* irrational algebraic functions
* exponential functions
* logarithmic functions
* hyperbolic functions
* inverse
* trigonometric functions
* inverse
* Secant
* Secant cubed

|

* List of limits
* List of derivatives

Miscellaneous topics|

* Complex calculus
* Contour integral
* Differential geometry
* Manifold
* Curvature
* of curves
* of surfaces
* Tensor
* Euler–Maclaurin formula
* Gabriel's horn
* Integration Bee
* Proof that 22/7 exceeds π
* Regiomontanus' angle maximization problem
* Steinmetz solid

* v
* t
* e

Major topics in mathematical analysis

* Calculus: Integration
* Differentiation
* Differential equations
* ordinary
* partial
* stochastic
* Fundamental theorem of calculus
* Calculus of variations
* Vector calculus
* Tensor calculus
* Matrix calculus
* Lists of integrals
* Table of derivatives

* Real analysis
* Complex analysis
* Hypercomplex analysis (quaternionic analysis)
* Functional analysis
* Fourier analysis
* Least-squares spectral analysis
* Harmonic analysis
* P-adic analysis (P-adic numbers)
* Measure theory
* Representation theory

* Functions
* Continuous function
* Special functions
* Limit
* Series
* Infinity

Mathematics portal
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