도함수표

1. 개요

2. 기본

2.1. 선형성(linearity)[1]

• [math(\left\{ \alpha f( x)+ \beta g(x) \right\}'=\alpha f'(x) +\beta g'( x ))]

2.2. 곱미분

• [math(\{f(x)g(x)\}'=f'(x)g(x)+f(x)g'(x))]
• [math(\{f(x)g(x)h(x)\}'=f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x))]

[math(\vdots)]

2.3. 몫미분

• [math(\left\{\dfrac{f(x)}{g(x)}\right\}'=\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x) ]^{2}} \; )]

2.4. 합성함수

• [math(\{(f\circ g)(x)\}'=(f' \circ g)(x)\,g'(x))]

2.5. 역함수

• [math(\{f^{-1}(x)\}'=\dfrac{1}{(f' \circ f^{-1})(x)})]

2.6. 정적분으로 정의된 함수

• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_a^xf(t)\,\mathrm{d}t = f(x))]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_x^af(t)\,\mathrm{d}t = -f(x))]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_a^{h(x)}f(t)\,\mathrm{d}t = (f\circ h)(x)\cdot h'(x))]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_{g(x)}^af(t)\,\mathrm{d}t = -(f\circ g)(x)\cdot g'(x))]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_{g(x)}^{h(x)}f(t)\,\mathrm{d}t = (f\circ h)(x)\cdot h'(x)-(f\circ g)(x)\cdot g'(x))]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_a^bf(x,\,t)\,\mathrm{d}t = \int_a^b \frac{\partial}{\partial x}f(x,\,t)\,\mathrm{d}t)]
• [math(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \int_{g(x)}^{h(x)}f(x,\,t)\,\mathrm{d}t = f(x,\,h(x))\cdot h'(x) - f(x,\,g(x))\cdot g'(x) + \int_{g(x)}^{h(x)}\frac{\partial}{\partial x}f(x,\,t)\,\mathrm{d}t )]

3. 초등함수

3.1. 다항함수

• [math((ax^n)'=anx^{n-1})] (단, [math(n \neq 0)]인 상수)
• 특히, [math(n=0)]이면
• [math(ax^0=a \; \to \; a'=0)] ( 상수함수)

3.2. 거듭제곱근 함수

• [math((\sqrt[n]{x})' = \dfrac{\sqrt[n]{x}}{nx})] (단, [math(n \neq 0)]인 상수)

3.3. 지수함수

• [math((a^{f(x)})'=a^{f(x)}f'(x)\cdot{\rm ln}\,a)] (단, [math(a)]는 상수)
• 특별히, [math(a=)] [math(e)]이면
• [math((e^{f(x)})'=e^{f(x)}f'(x)\cdot{\rm ln}\,e=e^{f(x)}f'(x))]
• 특별히, [math(f(x)=x)]이면
• [math((a^x)'=a^x\cdot{\rm ln}\,a)]
• 특별히, [math(a=)] [math(e)]이고 [math(f(x)=x)]이면
• [math((e^x)'=e^x\cdot {\rm ln}\,e=e^x)]

3.3.1. 루트

3.3.2. 허수지수함수

• [math([ \operatorname{cis}(z) ]' = i \operatorname{cis}(z))]
• [math([ \overline{\operatorname{cis}}(z) ]' = -i\,\overline{\operatorname{cis}}(z))]

3.4. 삼각함수

• [math(({\rm sin}\,x)'={\rm cos}\,x)]
• [math(({\rm cos}\,x)'=-{\rm sin}\,x)]
• [math(({\rm tan}\,x)'={\rm sec}^2\,x)]
• [math(({\rm cot}\,x)'=-{\rm csc^2}\,x)]
• [math(({\rm sec}\,x)'={\rm sec}\,x{\rm tan}\,x)]
• [math(({\rm csc}\,x)'=-{\rm csc}\,x{\rm cot}\,x)]

3.4.1. 싱크 함수

• [math(({\rm sinc}\,x)'= \dfrac{x \cos x - \sin x}{x^2})]

3.5. 역삼각함수

• [math((\arcsin x)' = \dfrac1{\sqrt{1-x^2}})]
• [math((\arccos x)' = -\dfrac1{\sqrt{1-x^2}})]
• [math((\arctan x)' = \dfrac1{1+x^2})]
• [math((\mathrm{arcsec}\,x)' = \dfrac1{\left|x\right|\sqrt{x^2-1}})]
• [math((\mathrm{arccsc}\,x)' = -\dfrac1{\left|x\right|\sqrt{x^2-1}})]
• [math((\mathrm{arccot}\,x)' = -\dfrac1{1+x^2})]

3.6. 쌍곡선 함수

• [math(({\rm sinh}\,x)'={\rm cosh}\,x)]
• [math(({\rm cosh}\,x)'={\rm sinh}\,x)]
• [math(({\rm tanh}\,x)'={\rm sech}^2\,x)]
• [math(({\rm coth}\,x)'=-{\rm csch}^2\,x)]
• [math(({\rm sech}\,x)'=-{\rm sech}\,x{\rm tanh}\,x)]
• [math(({\rm csch}\,x)'=-{\rm csch}\,x{\rm coth}\,x)]

3.7. 역쌍곡선 함수

• [math((\mathrm{arsinh}\,x)' = \dfrac1{\sqrt{x^2+1}})]
• [math((\mathrm{arcosh}\,x)' = \dfrac1{\sqrt{x^2-1}} \quad (x>1))]
• [math((\mathrm{artanh}\,x)' = \dfrac1{1-x^2} \quad (|x|<1))]
• [math((\mathrm{arcoth}\,x)' = \dfrac1{1-x^2} \quad (|x|>1))]
• [math((\mathrm{arsech}\,x)' = -\dfrac1{x\sqrt{1-x^2}} \quad (0<x<1))]
• [math((\mathrm{arcsch}\,x)' = -\dfrac1{|x|\sqrt{1+x^2}} \quad (x\ne0))]

3.8. 로그함수

• [math((\ln x)' = \dfrac{1}{x} \; (x > 0))]
• [math(({\rm Log}\,z)' = \dfrac{1}{z} \; (z \neq 0))]
• [math((\log_a x)' = \dfrac{1}{x \ln a} \; (x > 0))]

4. 특수함수

4.1. 오차함수

• [math([{\rm erf}(x) ]' = \dfrac{2e^{-x^2}}{\sqrt{\pi}})]
• [math([{\rm erfc}(x) ]' = -\dfrac{2e^{-x^2}}{\sqrt{\pi}})]
• [math([{\rm erfi}(x) ]' = \dfrac{2e^{x^2}}{\sqrt{\pi}})]

4.2. 지수 적분 함수

• [math([{\rm Ei}(x) ]' = \dfrac{e^x}{x})]

4.3. 로그 적분 함수

• [math([{\rm li}(x) ]' = \dfrac{1}{\ln x})]

4.4. 삼각 적분 함수

• [math([{\rm Si}(x) ]' = \dfrac{\sin x}{x})]
• [math([{\rm Ci}(x) ]' = \dfrac{\cos x}{x})]

4.5. 쌍곡선 적분 함수

• [math([{\rm Shi}(x) ]' = \dfrac{\sinh x}{x})]
• [math([{\rm Chi}(x) ]' = \dfrac{\cosh x}{x})]

4.6. 프레넬 적분 함수

• [math([S(x) ]' = \sin{ \!\left( \dfrac{\pi x^2}{2} \right) \, {\sf or} \; \sin x^2})]
• [math([C(x) ]' = \cos{ \!\left( \dfrac{\pi x^2}{2} \right) \, {\sf or} \; \cos x^2})]

4.7. 구데르만 함수

• [math([{\rm gd}(x) ]' = {\rm sech}\,x)]
• [math([{\rm igd}(x) ]' = \sec x)]

4.8. 타원 적분 함수

• [math(\displaystyle [K(x,\,k) ]' = \frac{1}{\sqrt{1-k^{2}}\sqrt{1-k^{2}x^{2} }} \quad (0 \leq k \leq 1))]
• [math(\displaystyle [E(x,\,k) ]' = \frac{\sqrt{1-k^{2}x^{2} }}{\sqrt{1-x^{2} }} \quad (0 \leq k \leq 1) )]

4.9. 브링 근호

• [math([\mathrm{BR}(x) ]' = -\dfrac{1}{5[ \mathrm{BR}(x) ]^4+1})]

4.10. 폴리로그함수

• [math([\mathrm{Li}_s(x) ]' = \dfrac{\mathrm{Li}_{s-1}(x)}{x})]

4.11. 제타 함수

• [math(\zeta'(s) = \displaystyle - \sum^{\infty}_{k=2} k^{-s} \ln k)]

4.12. 감마 함수

• [math(\Gamma'(z) = \Gamma(z) \psi(z) = \Gamma(z) [ {\rm Log}\,\Gamma(z) ]' )]

4.12.1. 로그 감마 함수

• [math([{\rm Log}\,\Gamma(z) ]' =\psi(z) = \dfrac{\Gamma'(z)}{\Gamma(z)} )]

4.13. 람베르트 W 함수

• [math(W'(x) = \dfrac1{e^{W(x)}+x})]

4.14. 부호 함수 ・ 헤비사이드 계단 함수

• [math([{\rm sgn}(x) ]' = 2 \delta(x))]
• [math(u'(x)= \delta(x))]

5. 음함수

5.1. 원

• [math((x-a)^2+(y-b)^2=r^2\;\rightarrow\;\dfrac{{\rm d}y}{{\rm d}x}=-\dfrac{x-a}{y-b})] ([math(a)], [math(b)], [math(r)]은 상수)

5.2. 타원곡선

• [math(y^2 = x^3 + ax + b\;\rightarrow\;\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3x^2 + a}{2y})] ([math(a)], [math(b)]는 상수)

6. 기타

• [math(\left(\displaystyle\int_0^xf(t)(x-t)\;{\rm d}t\right)'=\displaystyle\int_0^xf(t)\;{\rm d}t)]
• [math(|f(x)|' = ({\rm sgn} \circ f)(x)\,f'(x))]
• [math((x^x)'=x^x(1+{\rm ln}\,x))]
• [math(y=x^{x^{x^{x^{.^{.^.}} }} })][math(\!\!\rightarrow\; \dfrac{{\rm d}y}{{\rm d}x}=\dfrac{y^2}{(x-xy\ln{x})}=\dfrac{[W(-\operatorname{Log}{x}) ]^2}{ x \operatorname{Log}^{2}{x} [W(-\operatorname{Log}{x}) + 1 ] })]
  ([math(W(x))]는 람베르트 W 함수)

7. 관련 문서

• 역도함수표
• 미분
• 도함수
• 삼각함수/도함수