Rules for computing derivatives of functions
This article is a summary ofdifferentiation rules , that is, rules for computing thederivative of afunction incalculus .
Elementary rules of differentiation [ edit ] Unless otherwise stated, all functions are functions ofreal numbers (R {\textstyle \mathbb {R} } well defined ,[ 1] [ 2] complex numbers (C {\textstyle \mathbb {C} } [ 3]
For any value ofc {\textstyle c} c ∈ R {\textstyle c\in \mathbb {R} } f ( x ) {\textstyle f(x)} f ( x ) = c {\textstyle f(x)=c} d f d x = 0 {\textstyle {\frac {df}{dx}}=0} [ 4]
Letc ∈ R {\textstyle c\in \mathbb {R} } f ( x ) = c {\textstyle f(x)=c} 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.
[ edit ] Thederivative of the function at a point is the slope of the linetangent to the curve at the point. Theslope of the constant function is 0, because thetangent line to the constant function is horizontal and its angle is 0.
In other words, the value of the constant function,y {\textstyle y} x {\textstyle x}
At each point, thederivative is the slope of aline that istangent to thecurve at that point. Note: the derivative at point A ispositive where green and dash–dot,negative where red and dashed, and0 where black and solid. Differentiation is linear [ edit ] For any functionsf {\textstyle f} g {\textstyle g} a {\textstyle a} b {\textstyle b} h ( x ) = a f ( x ) + b g ( x ) {\textstyle h(x)=af(x)+bg(x)} x {\textstyle x} h ′ ( x ) = a f ′ ( x ) + b g ′ ( x ) {\textstyle h'(x)=af'(x)+bg'(x)}
InLeibniz'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',}
( f + g ) ′ = f ′ + g ′ , {\displaystyle (f+g)'=f'+g',}
( f − g ) ′ = f ′ − g ′ . {\displaystyle (f-g)'=f'-g'.}
For the functionsf {\textstyle f} g {\textstyle g} h ( x ) = f ( x ) g ( x ) {\textstyle h(x)=f(x)g(x)} x {\textstyle x} 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}}.}
The derivative of the functionh ( x ) = f ( g ( x ) ) {\textstyle h(x)=f(g(x))} 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),} 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 mapD {\textstyle {\text{D}}} [ 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 [ edit ] If the functionf {\textstyle f} inverse function g {\textstyle g} g ( f ( x ) ) = x {\textstyle g(f(x))=x} f ( g ( y ) ) = y {\textstyle f(g(y))=y} 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}}}.}
[ edit ] Polynomial or elementary power rule [ edit ] Iff ( x ) = x r {\textstyle f(x)=x^{r}} r ≠ 0 {\textstyle r\neq 0} f ′ ( x ) = r x r − 1 . {\displaystyle f'(x)=rx^{r-1}.}
Whenr = 1 {\textstyle r=1} f ( x ) = x {\textstyle f(x)=x} 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.
The derivative ofh ( x ) = 1 f ( x ) {\textstyle h(x)={\frac {1}{f(x)}}} f {\textstyle f} h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 , {\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},} f {\textstyle f}
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.
Iff {\textstyle f} g {\textstyle g} ( f g ) ′ = f ′ g − g ′ f g 2 , {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}},} g {\textstyle g}
This can be derived from the product rule and the reciprocal rule.
Generalized power rule [ edit ] The elementary power rule generalizes considerably. The most general power rule is thefunctional power rule : for any functionsf {\textstyle f} 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 }
Special cases:
Derivatives of exponential and logarithmic functions [ edit ] 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.} c {\displaystyle c} c < 0 {\displaystyle c<0}
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.} c {\textstyle c} 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},} W ( x ) {\textstyle W(x)} 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 [ edit ] Thelogarithmic derivative is another way of stating the rule for differentiating thelogarithm of a function (using the chain rule):( ln f ) ′ = f ′ f , {\displaystyle (\ln f)'={\frac {f'}{f}},} f {\textstyle f}
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 [ edit ] The derivatives in the table above are for when the range of the inverse secant is[ 0 , π ] {\textstyle [0,\pi ]} [ − π 2 , π 2 ] {\textstyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}
It is common to additionally define aninverse tangent function with two arguments ,arctan ( y , x ) {\textstyle \arctan(y,x)} [ − π , π ] {\textstyle [-\pi ,\pi ]} ( x , y ) {\textstyle (x,y)} x > 0 {\displaystyle x>0} arctan ( y , x > 0 ) = arctan ( y x ) {\textstyle \arctan(y,x>0)=\arctan({\frac {y}{x}})} ∂ 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 [ edit ] Derivatives of special functions [ edit ] 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}}} ψ ( x ) {\textstyle \psi (x)} digamma function , expressed by the parenthesized expression to the right ofΓ ( x ) {\textstyle \Gamma (x)}
Riemann zeta function [ edit ] 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 [ edit ] Suppose that it is required to differentiate with respect tox {\textstyle x} 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 functionsf ( x , t ) {\textstyle f(x,t)} ∂ ∂ x f ( x , t ) {\textstyle {\frac {\partial }{\partial x}}\,f(x,t)} t {\textstyle t} x {\textstyle x} ( t , x ) {\textstyle (t,x)} a ( x ) ≤ t ≤ b ( x ) {\textstyle a(x)\leq t\leq b(x)} x 0 ≤ x ≤ x 1 {\textstyle x_{0}\leq x\leq x_{1}} a ( x ) {\textstyle a(x)} b ( x ) {\textstyle b(x)} x 0 ≤ x ≤ x 1 {\textstyle x_{0}\leq x\leq x_{1}} 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 theLeibniz integral rule and can be derived using thefundamental theorem of calculus .
Derivatives ton th order [ edit ] Some rules exist for computing then {\textstyle n} n {\textstyle n}
[ edit ] Iff {\textstyle f} g {\textstyle g} n {\textstyle n} 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}},} r = ∑ m = 1 n − 1 k m {\textstyle r=\sum _{m=1}^{n-1}k_{m}} { k m } {\textstyle \{k_{m}\}} Diophantine equation ∑ m = 1 n m k m = n {\textstyle \sum _{m=1}^{n}mk_{m}=n}
General Leibniz rule [ edit ] Iff {\textstyle f} g {\textstyle g} n {\textstyle n} 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).}
^ Calculus (5th edition) , F. Ayres, E. Mendelson, Schaum's Outline Series, 2009,ISBN 978-0-07-150861-2 .^ Advanced Calculus (3rd edition) , R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010,ISBN 978-0-07-162366-7 .^ 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 ^ "Differentiation Rules" .University of Waterloo – CEMC Open Courseware . Retrieved3 May 2022 .Sources and further reading [ edit ] 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 .
