অন্তরকলন বিধি

এগুলো পৃথকীকরণের বিধিগুলোর একটি সংক্ষিপ্তসার। অর্থাৎ ক্যালকুলাসের যেকোন ক্রমের ডেরিভেটিভ গণনা করার নিয়ম। যদি f(x) একটি ধ্রুবক হয় , তাহলে f ′ = 0. {\displaystyle f'=0.\,} যোগের নিয়ম ( α f + β g ) ′ = α f ′ + β g ′ {\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'\,} α ও β বাস্তব সংখ্যা গুণের নিয়ম ( f g ) ′ = f ′ g + f g ′ {\displaystyle (fg)'=f'g+fg'\,} ভাগের নিয়ম ( f g ) ′ = f ′ g − f g ′ g 2 {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}  ;g ≠ 0 চেইন নিয়ম যদি f ( x ) = h ( g ( x ) ) {\displaystyle f(x)=h(g(x))} হয় তবে f ′ ( x ) = h ′ ( g ( x ) ) ⋅ g ′ ( x ) . {\displaystyle f'(x)=h'(g(x))\cdot g'(x).\,} d d x c = 0 {\displaystyle {d \over dx}c=0} d d x x = 1 {\displaystyle {d \over dx}x=1} d d x ( c x ) = c {\displaystyle {d \over dx}(cx)=c} d d x x n = n x n − 1 {\displaystyle {d \over dx}x^{n}=nx^{n-1}} যেখানে x n {\displaystyle x^{n}} ও n x n − 1 {\displaystyle nx^{n-1}} সংজ্ঞায়িত d d x c x = c x ln ⁡ c , c > 0 {\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0} d d x e x = e x {\displaystyle {d \over dx}e^{x}=e^{x}} d d x ln ⁡ x = 1 x , x > 0 {\displaystyle {d \over dx}\ln x={1 \over x},\qquad x>0} ] d d x sin ⁡ x = cos ⁡ x {\displaystyle {d \over dx}\sin x=\cos x} d d x cos ⁡ x = − sin ⁡ x {\displaystyle {d \over dx}\cos x=-\sin x} d d x tan ⁡ x = sec 2 ⁡ x = 1 cos 2 ⁡ x {\displaystyle {d \over dx}\tan x=\sec ^{2}x={1 \over \cos ^{2}x}} d d x sec ⁡ x = tan ⁡ x sec ⁡ x {\displaystyle {d \over dx}\sec x=\tan x\sec x} d d x sec ⁡ x = − sec ⁡ x cot ⁡ x {\displaystyle {d \over dx}\sec x=-\sec x\cot x} d d x cot ⁡ x = − sec 2 ⁡ x = − 1 sin 2 ⁡ x {\displaystyle {d \over dx}\cot x=-\sec ^{2}x={-1 \over \sin ^{2}x}} d d x arcsin ⁡ x = 1 1 − x 2 {\displaystyle {d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2}}}}} d d x arccos ⁡ x = − 1 1 − x 2 {\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}}}}} d d x arctan ⁡ x = 1 1 + x 2 {\displaystyle {d \over dx}\arctan x={1 \over 1+x^{2}}} d d x arcsec ⁡ x = 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}} d d x arccsc ⁡ x = − 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}} d d x arccot ⁡ x = − 1 1 + x 2 {\displaystyle {d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}} d d x sinh ⁡ x = cosh ⁡ x = e x + e − x 2 {\displaystyle {d \over dx}\sinh x=\cosh x={\frac {e^{x}+e^{-x}}{2}}} d d x cosh ⁡ x = sinh ⁡ x = e x − e − x 2 {\displaystyle {d \over dx}\cosh x=\sinh x={\frac {e^{x}-e^{-x}}{2}}} d d x tanh ⁡ x = sech 2 x {\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}\,x} d d x sech x = − tanh ⁡ x sech x {\displaystyle {d \over dx}\,\operatorname {sech} \,x=-\tanh x\,\operatorname {sech} \,x} d d x coth x = − csch 2 x {\displaystyle {d \over dx}\,\operatorname {coth} \,x=-\,\operatorname {csch} ^{2}\,x} d d x csch x = − coth x csch x {\displaystyle {d \over dx}\,\operatorname {csch} \,x=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x} d d x arcsinh x = 1 x 2 + 1 {\displaystyle {d \over dx}\,\operatorname {arcsinh} \,x={1 \over {\sqrt {x^{2}+1}}}} d d x arccosh x = 1 x 2 − 1 {\displaystyle {d \over dx}\,\operatorname {arccosh} \,x={1 \over {\sqrt {x^{2}-1}}}} d d x arctanh x = 1 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arctanh} \,x={1 \over 1-x^{2}}} d d x arcsech x = − 1 x 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arcsech} \,x={-1 \over x{\sqrt {1-x^{2}}}}} d d x arccoth x = 1 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arccoth} \,x={1 \over 1-x^{2}}} d d x arccsch x = − 1 | x | 1 + x 2 {\displaystyle {d \over dx}\,\operatorname {arccsch} \,x={-1 \over |x|{\sqrt {1+x^{2}}}}} d d x f ( c x ) = c f ′ ( c x ) {\displaystyle {d \over dx}f(cx)=cf'(cx)} d d x | x | = | x | x = sgn ⁡ x , x ≠ 0 {\displaystyle {d \over dx}|x|={|x| \over x}=\operatorname {sgn} x,\qquad x\neq 0} d d x ( f − 1 ( x ) ) = 1 f ′ ( f − 1 ( x ) ) {\displaystyle {d \over dx}(f^{-1}(x))={\frac {1}{f'(f^{-1}(x))}}} চেইন নিয়ম থেকে যা প্রমাণ করা যায়। অন্তরকলন সমাকলন সূচী