Diferensiasi fungsi trigonometri

Fungsi Turunan
sin ⁡ ( x ) {\displaystyle \sin(x)} cos ⁡ ( x ) {\displaystyle \cos(x)}
cos ⁡ ( x ) {\displaystyle \cos(x)} − sin ⁡ ( x ) {\displaystyle -\sin(x)}
tan ⁡ ( x ) {\displaystyle \tan(x)} sec 2 ⁡ ( x ) {\displaystyle \sec ^{2}(x)}
cot ⁡ ( x ) {\displaystyle \cot(x)} − csc 2 ⁡ ( x ) {\displaystyle -\csc ^{2}(x)}
sec ⁡ ( x ) {\displaystyle \sec(x)} sec ⁡ ( x ) tan ⁡ ( x ) {\displaystyle \sec(x)\tan(x)}
csc ⁡ ( x ) {\displaystyle \csc(x)} − csc ⁡ ( x ) cot ⁡ ( x ) {\displaystyle -\csc(x)\cot(x)}
arcsin ⁡ ( x ) {\displaystyle \arcsin(x)} 1 1 − x 2 {\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}
arccos ⁡ ( x ) {\displaystyle \arccos(x)} − 1 1 − x 2 {\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}
arctan ⁡ ( x ) {\displaystyle \arctan(x)} 1 x 2 + 1 {\displaystyle {\frac {1}{x^{2}+1}}}

Diferensiasi fungsi trigonometri atau turunan fungsi trigonometri adalah proses matematis untuk menemukan turunan suatu fungsi trigonometri atau tingkat perubahan terkait dengan suatu variabelnya. Fungsi trigonometri yang umum digunakan adalah sin(x), cos(x) dan tan(x). Contohnya, turunan "f(x) = sin(x)" dituliskan "f ′(a) = cos(a)". "f ′(a)" adalah tingkat perubahan sin(x) di titik "a".

Semua turunan fungsi trigonometri lingkaran dapat ditemukan dengan menggunakan turunan sin(x) dan cos(x). Kaidah hasil-bagi lalu digunakan untuk menemukan turunannya. Sementara itu, pencarian turunan fungsi trigonometri invers membutuhkan diferensiasi implisit dan turunan fungsi trigonometri biasa.

Turunan fungsi trigonometri

d d x sin ⁡ ( x ) = cos ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}
d d x cos ⁡ ( x ) = − sin ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}
d d x tan ⁡ ( x ) = ( sin ⁡ ( x ) cos ⁡ ( x ) ) ′ = cos 2 ⁡ ( x ) + sin 2 ⁡ ( x ) cos 2 ⁡ ( x ) = 1 + tan 2 ⁡ ( x ) = sec 2 ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\tan(x)=\left({\frac {\sin(x)}{\cos(x)}}\right)'={\frac {\cos ^{2}(x)+\sin ^{2}(x)}{\cos ^{2}(x)}}=1+\tan ^{2}(x)=\sec ^{2}(x)}
d d x cot ⁡ ( x ) = ( cos ⁡ ( x ) sin ⁡ ( x ) ) ′ = − sin 2 ⁡ ( x ) − cos 2 ⁡ ( x ) sin 2 ⁡ ( x ) = − ( 1 + cot 2 ⁡ ( x ) ) = − csc 2 ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\cot(x)=\left({\frac {\cos(x)}{\sin(x)}}\right)'={\frac {-\sin ^{2}(x)-\cos ^{2}(x)}{\sin ^{2}(x)}}=-(1+\cot ^{2}(x))=-\csc ^{2}(x)}
d d x sec ⁡ ( x ) = ( 1 cos ⁡ ( x ) ) ′ = sin ⁡ ( x ) cos 2 ⁡ ( x ) = 1 cos ⁡ ( x ) ⋅ sin ⁡ ( x ) cos ⁡ ( x ) = sec ⁡ ( x ) tan ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\sec(x)=\left({\frac {1}{\cos(x)}}\right)'={\frac {\sin(x)}{\cos ^{2}(x)}}={\frac {1}{\cos(x)}}\cdot {\frac {\sin(x)}{\cos(x)}}=\sec(x)\tan(x)}
d d x csc ⁡ ( x ) = ( 1 sin ⁡ ( x ) ) ′ = − cos ⁡ ( x ) sin 2 ⁡ ( x ) = − 1 sin ⁡ ( x ) ⋅ cos ⁡ ( x ) sin ⁡ ( x ) = − csc ⁡ ( x ) cot ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\csc(x)=\left({\frac {1}{\sin(x)}}\right)'=-{\frac {\cos(x)}{\sin ^{2}(x)}}=-{\frac {1}{\sin(x)}}\cdot {\frac {\cos(x)}{\sin(x)}}=-\csc(x)\cot(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 arccos ⁡ ( x ) = − 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arccos(x)={\frac {-1}{\sqrt {1-x^{2}}}}}
d d x arctan ⁡ ( x ) = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}
d d x arccot ( x ) = − 1 1 + x 2 {\displaystyle {\frac {d}{dx}}{\mbox{arccot}}(x)={\frac {-1}{1+x^{2}}}}
d d x arcsec ( x ) = 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}{\mbox{arcsec}}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x arccsc ( x ) = − 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}{\mbox{arccsc}}(x)={\frac {-1}{|x|{\sqrt {x^{2}-1}}}}}

Daftar pustaka

  • Handbook of Mathematical Functions, Edited by Abramowitz and Stegun, National Bureau of Standards, Applied Mathematics Series, 55 (1964)