This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Kaidah pendiferensialan (atau aturan pendiferensialan; bahasa Inggris: Rules of differentiationcode: en is deprecated ) berikut merupakan ringkasan kaidah-kaidah untuk menghitung derivatif suatu fungsi dalam kalkulus.
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, including the case of complex numbers ( C {\textstyle \mathbb {C} } ).
Kecuali dinyatakan lain, semua fungsi merupakan fungsi bilangan real (R) yang menghasilkan nilai bilangan real; meskipun secara lebih umum, rumus-rumus berikut dapat diterapkan di manapun jika didefinisikan dengan baik— termasuk bilangan kompleks (C).
For the functions f {\textstyle f} and g {\textstyle g} , the derivative of the function h ( x )f ( x )
Untuk fungsi-fungsi f dan g, turunan fungsi h(x)f(x) g(x) terhadap x dapat ditulis h ′ ( x )
( f g ) ′ ( x )f ′ ( x )
f ′ ( g ( x ) )
d xg d f d x + f d g d x .
d xd f d x g + f d g d x .
{dx}}=g{\frac {df}{dx}}+f{\frac {dg}{dx}}.} The derivative of the function h ( x )f ( g ( x ) ) {\textstyle h(x)=f(g(x))} is: h ′ ( x )
{dx}}={\frac {dh}{dg}}{\frac {dg}{dx}}.\,} Jika fungsi f mempunyai suatu fungsi invers g, yaitu g(f(x))
f ′ ( g ( x ) ) ⋅ g ′ ( x ) .
f ′ ( g ( x ) )
d d z f ( z ) | zg ( x ) ⋅ d d x g ( x ) , {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 d x ( ln ⁡ x )1 x , x > 0 {dx}}\left(\ln x\right)={1 \over x},\qquad x>0} d d x ( ln ⁡ | x | )1 x {dx}}\left(\ln |x|\right)={1 \over x}} d d x ( x x )x x ( 1 + ln ⁡ x ) .
d xd f ( g ( x ) )
d h d xd f ( g ( x ) )
d x . {dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}
d x . {dy}}={\frac {1}{dy/dx}}.}
[ D f ] g ( x ) ⋅ [ D g ] x . }(f\circ g)]_{x}=[{\text{D}}f]_{g(x)}\cdot
{dx}}={\frac {dh}{dg}}{\frac {dg}{dx}}.\,} Jika fungsi f mempunyai suatu fungsi invers g, yaitu g(f(x))
If the function f {\textstyle f} has an inverse function g {\textstyle g} , meaning that g ( f ( x ) )
{dx}}={\frac {dh}{dg}}{\frac {dg}{dx}}.\,} Jika fungsi f mempunyai suatu fungsi invers g, yaitu g(f(x))
1 f ( x ) {\textstyle h(x)={\frac {1}{f(x)}}} for any (nonvanishing) function f {\textstyle f} is: h ′ ( x )
Untuk fungsi-fungsi f dan g, turunan fungsi h(x)f(x) g(x) terhadap x dapat ditulis h ′ ( x )
d x− 1 f 2 d f d x .
d xa d f d x + b d g d x .
{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.
d x− 1 f 2 d f d x . {dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.\,} Kaidah timbal balik (reciprocal rule) dapat diturunkan dari kaidah rantai (chain rule) dan kaidah pemangkatan (kaidah pangkat; power rule).
If f {\textstyle f} and g {\textstyle g} are functions, then: ( f g ) ′f ′ g − g ′ f g 2 , {g}}\right)'={\frac {f'g-g'f}{g^{2}}},} wherever g {\textstyle g} is nonzero.
Jika f dan g adalah fungsi, maka: ( f g ) ′f ′ g − g ′ f g 2 {g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad } di mana g bukan nol.
d d x ( e a x )a e a x .
d xa d f d x + b d g d x .
{dx}}\left(e^{ax}\right)=ae^{ax}.} d d x ( log c ⁡ x )1 x ln ⁡ c , c > 1. {dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1.}
d d x ( e x )e x {dx}}\left(e^{x}\right)=e^{x}} d d x ( log c ⁡ x )1 x ln ⁡ c , c > 0 , c ≠ 1 {dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1} persamaan di atas adalah benar untuk semua c, tetapi turunan bagi c < 0 menghasilkan bilangan kompleks.
1 x , x ≠ 0. {dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.}
Maka untuk x 0 ≤ x ≤ x 1 \leq x\leq x_{1}\,\,} : F ′ ( x )
d d x ( W ( x ) )1 x + e W ( x ) , x > − 1 e , {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 ( ln ⁡ x )1 x , x > 0 {dx}}\left(\ln x\right)={1 \over x},\qquad x>0} d d x ( ln ⁡ | x | )1 x {dx}}\left(\ln |x|\right)={1 \over x}} d d x ( x x )x x ( 1 + ln ⁡ x ) .
d d x ( x x )x x ( 1 + ln ⁡ x ) .
d xa d f d x + b d g d x .
{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).} d d x ( f 1 ( x ) f 2 ( x ) ( . . . ) f n ( x ) )[ ∑ k1 n ∂ ∂ x k ( f 1 ( x 1 ) f 2 ( x 2 ) ( . . . ) f n ( x n ) ) ] | x 1x 2. . .
d d x ( ln ⁡ x )1 x , x > 0 {dx}}\left(\ln x\right)={1 \over x},\qquad x>0} d d x ( ln ⁡ | x | )1 x {dx}}\left(\ln |x|\right)={1 \over x}} d d x ( x x )x x ( 1 + ln ⁡ x ) .
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): ( ln ⁡ f )
Turunan logaritmik adalah cara lain untuk menyatakan kaidah diferensiasi logaritma suatu fungsi (menggunakan kaidah rantai): ( ln ⁡ f ) ′f ′ f {f}}\quad } wherever f is positive.
[ − π , π ] {\textstyle [-\pi ,\pi ]} and reflects the quadrant of the point ( x , y ) {\textstyle (x,y)} .
[ − π , π ] dan mencerminkan kuadran dari titik ( x , y ) .
For the first and fourth quadrant (i.e., x > 0 ), one has arctan ⁡ ( y ,
Untuk kuadran pertama dan keempat (yaitu x > 0 ) maka arctan ⁡ ( y , x > 0 )arctan ⁡ ( y / x ) .
arctan ⁡ ( y x ) {\textstyle \arctan(y,x>0)=\arctan({\frac {y}{x}})} .
Untuk kuadran pertama dan keempat (yaitu x > 0 ) maka arctan ⁡ ( y , x > 0 )arctan ⁡ ( y / x ) .
d t , ^{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}} .
d t , ^{b(x)}f(x,t)\,dt,} di mana fungsi-fungsi f ( x , t ) dan ∂ ∂ x f ( x , t ) {\partial x}}\,f(x,t)\,} keduanya kontinu dalam t dan x dalam wilayah tertentu bidang ( t , x ) , termasuk a ( x ) ≤ t ≤ b ( x ) , x 0 ≤ x ≤ x 1 \leq x\leq x_{1}\,} , dan fungsi-fungsi a ( x ) dan b ( x ) keduanya kontinu dan memiliki turunan kontinu
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.
Rumus ini merupakan bentuk umum dari kaidah integral Leibniz dan dapat diturunkan menggunakan Teorema fundamental kalkulus.
d n d x n [ f ( g ( x ) ) ]n !
d n − k d x n − k f ( x )
k m , }{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∑ m1 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 ∑ m1 n m k mn {\textstyle \sum _{m=1}^{n}mk_{m}=n} .
k m }{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}}} di mana r∑ m1 n − 1 k m ^{n-1}k_{m}} dan himpunan { k m } \}} terdiri dari semua solusi bilangan bulat bukan negatif dari persamaan Diophantine ∑ m1 n m k mn ^{n}mk_{m}=n} .
d n d x n [ f ( x )
d n − k d x n − k f ( x )
d k d x k g ( x ) .
d n − k d x n − k f ( x )
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.
Notasi dalam halaman ini (selain pada rujukan-rujukan di atas) dapat dijumpai dalam: Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISBN 978-0-07-154855-7.