( x α ) ′ = α x α (x ^ \alpha)' = \alpha x ^ \alpha (xα)′=αxα
( sin x ) ′ = cos x (\sin x)' = \cos x (sinx)′=cosx
( cos x ) ′ = sin x (\cos x)' = \sin x (cosx)′=sinx
( tan x ) ′ = sec 2 x (\tan x)' = \sec ^ 2 x (tanx)′=sec2x
( cot x ) ′ = − csc 2 x (\cot x)' = -\csc ^ 2 x (cotx)′=−csc2x
( sec x ) ′ = sec x ⋅ tan x (\sec x)' = \sec x \cdot \tan x (secx)′=secx⋅tanx
( csc x ) ′ = − csc x ⋅ cot x (\csc x)' = -\csc x \cdot \cot x (cscx)′=−cscx⋅cotx
( arcsin x ) ′ = 1 1 − x 2 (\arcsin x)' = \frac{1}{\sqrt{1 - x ^ 2}} (arcsinx)′=1−x21
( arccos x ) ′ = − 1 1 − x 2 (\arccos x)' = -\frac{1}{\sqrt{1 - x ^ 2}} (arccosx)′=−1−x21
( arctan x ) ′ = 1 1 + x 2 (\arctan x)' = \frac{1}{1 + x ^ 2} (arctanx)′=1+x21
( a r c c o t x ) ′ = − 1 1 + x 2 (arccot \ x)' = - \frac{1}{1 + x ^ 2} (arccot x)′=−1+x21
( a x ) ′ = a x ⋅ ln a (a ^ x)' = a ^ x \cdot \ln a (ax)′=ax⋅lna
( log a x ) ′ = 1 x ⋅ ln a (\log_a x)' = \frac{1}{x \cdot \ln a} (logax)′=x⋅lna1
C ′ = 0 C' = 0 C′=0
( a x ) ( n ) = a x ⋅ ( ln a ) ( n ) (a ^ x) ^ {(n)} = a ^ x \cdot (\ln a) ^ {(n)} (ax)(n)=ax⋅(lna)(n)
( sin x ) ( n ) = sin ( x + n π 2 ) (\sin x) ^ {(n)} = \sin (x + \frac{n \pi}{2}) (sinx)(n)=sin(x+2nπ)
( cos x ) ( n ) = cos ( x + n π 2 ) (\cos x) ^ {(n)} = \cos (x + \frac{n \pi}{2}) (cosx)(n)=cos(x+2nπ)
[ ln ( 1 + x ) ] ( n ) = ( − 1 ) ( n − 1 ) ( n − 1 ) ! ( 1 + x ) ( n ) [\ln (1 + x)] ^ {(n)} = (-1) ^ {(n-1)} \frac{(n-1)!}{(1+x) ^ {(n)}} [ln(1+x)](n)=(−1)(n−1)(1+x)(n)(n−1)!
( 1 1 + x ) ( n ) = ( − 1 ) ( n ) ( n ) ! ( 1 + x ) ( n + 1 ) (\frac {1}{1+x}) ^ {(n)} = (-1) ^ {(n)} \frac{(n)!}{(1+x) ^ {(n+1)}} (1+x1)(n)=(−1)(n)(1+x)(n+1)(n)!
( x α ) ( n ) = α ( α − 1 ) . . . ( α − n + 1 ) x ( α − n ) (x ^ \alpha) ^ {(n)} = \alpha (\alpha -1)...(\alpha - n + 1) x ^ {(\alpha - n)} (xα)(n)=α(α−1)...(α−n+1)x(α−n)
∫ x α d x = 1 α + 1 x α + 1 + C ∫ 1 x d x = ln ∣ x ∣ + C ∫ a x d x = a x ln a + C ∫ e x d x = e x + C ∫ sin x d x = − cos x + C ∫ cos x d x = sin x + C ∫ tan x d x = − ln ∣ cos x ∣ + C ∫ cot x d x = ln ∣ sin x ∣ + C ∫ sec x d x = ln ∣ sec x + tan x ∣ + C ∫ csc x d x = ln ∣ csc x − cot x ∣ + C ∫ sec x tan x d x = sec x + C ∫ csc x cot x d x = − csc x + C ∫ sec 2 x d x = tan x + C ∫ csc 2 x d x = − cot x + C ∫ 1 x 2 + a 2 d x = 1 a arctan x a + C ∫ 1 x 2 − a 2 d x = 1 2 a ln ∣ x − a x + a ∣ + C ∫ 1 a 2 − x 2 d x = arcsin x a + C ∫ 1 x 2 + a 2 d x = ln ∣ x + x 2 + a 2 ∣ + C ∫ 1 x 2 − a 2 d x = ln ∣ x + x 2 − a 2 ∣ + C ∫ sinh x d x = cosh x + C ∫ cosh x d x = sinh x + C ∫ 1 sinh 2 x d x = − coth x + C ∫ 1 cosh 2 x d x = tanh x + C \int x ^ \alpha dx = \frac{1}{\alpha + 1} x ^ {\alpha + 1} + C \\\\ \int \frac{1}{x} dx = \ln |x| + C \\\\ \int a ^ x dx = \frac{a ^ x}{\ln a} + C \\\\ \int e ^ x dx = e ^ x + C \\\\ \int \sin x dx = -\cos x + C \\\\ \int \cos x dx = \sin x + C \\\\ \int \tan x dx = -\ln |\cos x| + C \\\\ \int \cot x dx = \ln |\sin x| + C \\\\ \int \sec x dx = \ln |\sec x + \tan x| + C \\\\ \int \csc x dx = \ln |\csc x - \cot x| + C \\\\ \int \sec x \tan x dx = \sec x + C \\\\ \int \csc x \cot x dx = -\csc x + C \\\\ \int \sec ^ 2 x dx = \tan x + C \\\\ \int \csc ^ 2 x dx = -\cot x + C \\\\ \int \frac{1}{x ^ 2 + a ^ 2} dx = \frac{1}{a} \arctan \frac{x}{a} + C \\\\ \int \frac{1}{x ^ 2 - a ^ 2} dx = \frac{1}{2a} \ln |\frac{x - a}{x + a}| + C \\\\ \int \frac{1}{\sqrt{a ^ 2 - x ^ 2}} dx = \arcsin \frac{x}{a} + C \\\\ \int \frac{1}{\sqrt{x ^ 2 + a ^ 2}} dx = \ln |x + \sqrt{x ^ 2 + a ^ 2}| + C \\\\ \int \frac{1}{\sqrt{x ^ 2 - a ^ 2}} dx = \ln |x + \sqrt{x ^ 2 - a ^ 2}| + C \\\\ \int \sinh x dx = \cosh x + C \\\\ \int \cosh x dx = \sinh x + C \\\\ \int \frac{1}{\sinh ^ 2 x} dx = -\coth x + C \\\\ \int \frac{1}{\cosh ^ 2 x} dx = \tanh x + C ∫xαdx=α+11xα+1+C∫x1dx=ln∣x∣+C∫axdx=lnaax+C∫exdx=ex+C∫sinxdx=−cosx+C∫cosxdx=sinx+C∫tanxdx=−ln∣cosx∣+C∫cotxdx=ln∣sinx∣+C∫secxdx=ln∣secx+tanx∣+C∫cscxdx=ln∣cscx−cotx∣+C∫secxtanxdx=secx+C∫cscxcotxdx=−cscx+C∫sec2xdx=tanx+C∫csc2xdx=−cotx+C∫x2+a21dx=a1arctanax+C∫x2−a21dx=2a1ln∣x+ax−a∣+C∫a2−x21dx=arcsinax+C∫x2+a21dx=ln∣x+x2+a2∣+C∫x2−a21dx=ln∣x+x2−a2∣+C∫sinhxdx=coshx+C∫coshxdx=sinhx+C∫sinh2x1dx=−cothx+C∫cosh2x1dx=tanhx+C
sin x ∼ x ∼ tan x arcsin x ∼ x ∼ arctan x 1 − cos x ∼ 1 2 x 2 1 + x n − 1 ∼ 1 n x ln ( 1 + x ) ∼ x e x − 1 ∼ x ( 1 + x ) α − 1 ∼ α x sinh x ∼ sin x ∼ x 1 + x − 1 − x ∼ x \sin x \sim x \sim \tan x \\\\ \arcsin x \sim x \sim \arctan x \\\\ 1 - \cos x \sim \frac{1}{2}x ^ 2 \\\\ \sqrt[n]{1 + x} - 1 \sim \frac{1}{n} x \\\\ \ln (1 + x) \sim x \\\\ e ^ x - 1 \sim x \\\\ (1 + x) ^ \alpha - 1 \sim \alpha x \\\\ \sinh x \sim \sin x \sim x \\\\ \sqrt{1 + x} - \sqrt{1 - x} \sim x sinx∼x∼tanxarcsinx∼x∼arctanx1−cosx∼21x2n1+x−1∼n1xln(1+x)∼xex−1∼x(1+x)α−1∼αxsinhx∼sinx∼x1+x−1−x∼x