উইকিপিডিয়া, মুক্ত বিশ্বকোষ থেকে
অন্তরকলনযোগ্য ফাংশন
এগুলো পৃথকীকরণের বিধিগুলোর একটি সংক্ষিপ্তসার। অর্থাৎ ক্যালকুলাসের যেকোন ক্রমের ডেরিভেটিভ গণনা করার নিয়ম।
- যদি f(x) একটি ধ্রুবক হয় , তাহলে
![{\displaystyle f'=0.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2834902ce1c3b0605dd8c631c0e7b1d904fdc5f5)
α ও β বাস্তব সংখ্যা
![{\displaystyle (fg)'=f'g+fg'\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f2ca6050d6691d537f9fafeea1a5f79e9d7cee)
;g ≠ 0
যদি
হয় তবে
![{\displaystyle f'(x)=h'(g(x))\cdot g'(x).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85dcc7c195a2660e8d2620b7b9d838a83575edbf)
মূলদ অপেক্ষকের সূত্র[সম্পাদনা]
![{\displaystyle {d \over dx}c=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d66384c0fe25d80d440b317f08c9b8be9253e77)
![{\displaystyle {d \over dx}x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03e3d9ed50d0216e5b16c4827596e3fdcc2deacb)
![{\displaystyle {d \over dx}(cx)=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1da6def131ad3ace3b1cb7369eac0bedc8d84be)
যেখানে
ও
সংজ্ঞায়িত
![{\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d595b56138acbd74c00aae36d407fe71f3b2dd7c)
![{\displaystyle {d \over dx}e^{x}=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2528d3e349a763e4afd73cddc3ec599ffca15e4)
লগ্যারিদমিক সূত্র[সম্পাদনা]
]
ত্রিকোণামিতিক সূত্র[সম্পাদনা]
![{\displaystyle {d \over dx}\sin x=\cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c976bcdb6b29573bc73f234a44acd4f9bb3ea8d3)
![{\displaystyle {d \over dx}\cos x=-\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbeeef651fc6d4f1515c348981a99f97e5d9ab3)
![{\displaystyle {d \over dx}\tan x=\sec ^{2}x={1 \over \cos ^{2}x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eecb39df5f14e2474ae267284a6e0147f5edfb33)
![{\displaystyle {d \over dx}\sec x=\tan x\sec x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f114356317ff48490ac235e8d69edb95e564547)
![{\displaystyle {d \over dx}\sec x=-\sec x\cot x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5a157ca1bda213b38a622a14c668f283232f85)
![{\displaystyle {d \over dx}\cot x=-\sec ^{2}x={-1 \over \sin ^{2}x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac2d3c30dd9d35738130584393d9b3919ab7209)
বিপরীত ত্রিকোণামিতিক সূত্র[সম্পাদনা]
![{\displaystyle {d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3695240e676e214b65661196c477a8d06ad25145)
![{\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/694f1f1ec9823b1af7a76c435f1be1b8150536a4)
![{\displaystyle {d \over dx}\arctan x={1 \over 1+x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ffbeab56eb8ceb4a7bd93e7341b6a45170ba0ab)
![{\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42da001ad7ea996603263d152a9479be330dafd3)
![{\displaystyle {d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15a59301aa2b9bd7d3c24012712d119ae367bf4b)
![{\displaystyle {d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e67da0f756cdcf3ace4c4f2fc64f822b4b5a40d)
হাইপারবোলিক সূত্র[সম্পাদনা]
![{\displaystyle {d \over dx}\sinh x=\cosh x={\frac {e^{x}+e^{-x}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd585210df2d572ee69b2d0decca68d60532641)
![{\displaystyle {d \over dx}\cosh x=\sinh x={\frac {e^{x}-e^{-x}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e82be1ab0ae8912f87d0edb4ad40ed3e0fe6a8)
![{\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}\,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a56722a21abd43b5eeddcc7744d8c31e97ae539)
![{\displaystyle {d \over dx}\,\operatorname {sech} \,x=-\tanh x\,\operatorname {sech} \,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28880630ee4e456ecbc143a9b123e4705eeb42e3)
![{\displaystyle {d \over dx}\,\operatorname {coth} \,x=-\,\operatorname {csch} ^{2}\,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64ceb55e3a64e4aba52689b1be832a22e2900f16)
![{\displaystyle {d \over dx}\,\operatorname {csch} \,x=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/238fb46cafcce56060527323df58700a51db119a)
বিপরীত হাইপারবোলিক সূত্র[সম্পাদনা]
![{\displaystyle {d \over dx}\,\operatorname {arcsinh} \,x={1 \over {\sqrt {x^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789104eecbf1e0a1862d0a8a54756fd4c92c967a)
![{\displaystyle {d \over dx}\,\operatorname {arccosh} \,x={1 \over {\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4c92e415b1105ce4f0520e0c0e8ed59a159a47)
![{\displaystyle {d \over dx}\,\operatorname {arctanh} \,x={1 \over 1-x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fcf6ba7ee2f290bc1d1ad13386ad085634358cb)
![{\displaystyle {d \over dx}\,\operatorname {arcsech} \,x={-1 \over x{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e04b2d1c4c3fff12a7878bd16dcd1494ed2d8d)
![{\displaystyle {d \over dx}\,\operatorname {arccoth} \,x={1 \over 1-x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f67ea12ad2b5bb503f08fb71d1a0879adb6ce3)
![{\displaystyle {d \over dx}\,\operatorname {arccsch} \,x={-1 \over |x|{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b42282ca2264d3622eb50d9e6db94d8edfafc86f)
![{\displaystyle {d \over dx}f(cx)=cf'(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6940f2a00f42475e605c36d66684fc45663cd2db)
![{\displaystyle {d \over dx}|x|={|x| \over x}=\operatorname {sgn} x,\qquad x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350ec1c5235e07d0aaa3ffadbf597273a1caf291)
চেইন নিয়ম থেকে যা প্রমাণ করা যায়।
বিশেষ অন্তরজ ফাংশন[সম্পাদনা]
- রাইমান যেটা (Zeta)ফাংশন
![{\displaystyle \quad \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4ac800cedd3a1c16350c75b4d85f1a867879b8)
![{\displaystyle \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 \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d6adc3cabac2f040a5af752e79ec4fb0a9bd18)
![{\displaystyle \,=-\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}}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ba698824ff4141d648906a853b286573a80608)
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