파생상품 치트 시트

파생상품 규칙

전원 규칙 \frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1}

상수의 도함수 \frac{d}{dx}\left(a\right)=0

합계 차이 규칙 \left(f\pm g\right)^'=f^'\pm g^'

상수 출력 \left(a\cdot f\right)^'=a\cdot f^'

제품 규칙 (f\cdot g)^'=f^'\cdot g+f\cdot g^'

몫 규칙 \left(\frac{f}{g}\right)^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}

연쇄 법칙 \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}

\frac{d}{dx}\left(\ln(x))=\frac{1}{x} \frac{d}{dx}\left(\ln(\left|x\right|))=\frac{1}{x}

\frac{d}{dx}\left(e^{x})=e^{x} \frac{d}{dx}\left(\log(x))=\frac{1}{x\ln(10)}

\frac{d}{dx}\left(\log_{a}(x))=\frac{1}{x\ln(a)}

\frac{d}{dx}\left(\sin(x))=\cos(x) \frac{d}{dx}\left(\cos(x))=-\sin(x)

\frac{d}{dx}\left(\tan(x))=\sec^{2}(x) \frac{d}{dx}\left(\sec(x))=\frac{\tan(x)}{\cos(x)}

\frac{d}{dx}\left(\csc(x))=\frac{-\cot(x)}{\sin(x)} \frac{d}{dx}\left(\cot(x))=-\frac{1}{\sin^{2}(x)}

\frac{d}{dx}\left(\arcsin(x))=\frac{1}{\sqrt{1-x^{2}}} \frac{d}{dx}\left(\arccos(x))=-\frac{1}{\sqrt{1-x^{2}}}

\frac{d}{dx}\left(\arctan(x))=\frac{1}{x^{2}+1} \frac{d}{dx}\left(\arcsec(x))=\frac{1}{\sqrt{x^2}\sqrt{x^2-1}}

\frac{d}{dx}\left(\arccsc(x))=-\frac{1}{\sqrt{x^2}\sqrt{x^2-1}} \frac{d}{dx}\left(\arccot(x))=-\frac{1}{x^{2}+1}

\frac{d}{dx}\left(\sinh(x))=\cosh(x) \frac{d}{dx}\left(\cosh(x))=\sinh(x)

\frac{d}{dx}\left(\tanh(x))=\sech^{2}(x) \frac{d}{dx}\left(\sech(x))=\tanh(x)(-\sech(x))

\frac{d}{dx}\left(\csch(x))=-\coth(x)\csch(x) \frac{d}{dx}\left(\coth(x))=-\csch^{2}(x)

\frac{d}{dx}\left(\arcsinh(x))=\frac{1}{\sqrt{x^{2}+1}} \frac{d}{dx}\left(\arccosh(x))=\frac{1}{\sqrt{x-1}\sqrt{x+1}}

\frac{d}{dx}\left(\arctanh(x))=\frac{1}{1-x^2} \frac{d}{dx}\left(\arcsech(x))=\frac{\sqrt{\frac{2}{x+1}-1}}{(x-1)x}

\frac{d}{dx}\left(\arccsch(x))=-\frac{1}{\sqrt{\frac{1}{x^2}+1}x^2} \frac{d}{dx}\left(\arccoth(x))=\frac{1}{1-x^{2}}