Simulab Apuntes

Límites, Derivadas e Integrales

Límites

LímiteResultado
lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}1
lim⁡x→0xsin⁡x\lim_{x \to 0} \frac{x}{\sin x}1
lim⁡x→0tan⁡xx\lim_{x \to 0} \frac{\tan x}{x}1
lim⁡x→∞(1+1x)x\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^xee
lim⁡x→∞(1+kx)x\lim_{x \to \infty} \left( 1 + \frac{k}{x} \right)^xeke^k
lim⁡x→∞(1+kx+a)x+a\lim_{x \to \infty} \left( 1 + \frac{k}{x + a} \right)^{x+a}eke^k

Reglas de Derivación

ReglaResultado
y=f(x)g(x)y = f(x)g(x)y′=f′(x)g(x)+f(x)g′(x)y' = f'(x)g(x) + f(x)g'(x)
y=f(x)g(x)y = \frac{f(x)}{g(x)}y′=f′(x)g(x)−f(x)g′(x)g2(x)y' = \frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}
h(x)=f(g(x))h(x) = f(g(x))h′(x)=f′(g(x))⋅g′(x)h'(x) = f'(g(x)) \cdot g'(x)

Derivadas

FunciónDerivada
f(x)=kf(x) = kf′(x)=0f'(x) = 0
f(x)=xf(x) = xf′(x)=1f'(x) = 1
f(x)=cxf(x) = cxf′(x)=cf'(x) = c
f(x)=xnf(x) = x^nf′(x)=n⋅xn−1f'(x) = n \cdot x^{n-1}
f(x)=ln⁡xf(x) = \ln xf′(x)=1xf'(x) = \frac{1}{x}
f(x)=sin⁡xf(x) = \sin xf′(x)=cos⁡xf'(x) = \cos x
f(x)=cos⁡xf(x) = \cos xf′(x)=−sin⁡xf'(x) = -\sin x
f(x)=exf(x) = e^xf′(x)=exf'(x) = e^x
f(x)=log⁡axf(x) = \log_a xf′(x)=1xln⁡af'(x) = \frac{1}{x \ln a}
f(x)=tan⁡xf(x) = \tan xf′(x)=sec⁡2xf'(x) = \sec^2 x
f(x)=cot⁡xf(x) = \cot xf′(x)=−csc⁡2xf'(x) = -\csc^2 x
f(x)=sec⁡xf(x) = \sec xf′(x)=sec⁡xtan⁡xf'(x) = \sec x \tan x
f(x)=csc⁡xf(x) = \csc xf′(x)=−csc⁡xcot⁡xf'(x) = -\csc x \cot x
f(x)=axf(x) = a^xf′(x)=axln⁡af'(x) = a^x \ln a
f(x)=tan⁡−1xf(x) = \tan^{-1} xf′(x)=11+x2f'(x) = \frac{1}{1 + x^2}
f(x)=sin⁡−1xf(x) = \sin^{-1} xf′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}}
f(x)=cos⁡−1xf(x) = \cos^{-1} xf′(x)=−11−x2f'(x) = \frac{-1}{\sqrt{1 - x^2}}
f(x)=sinh⁡xf(x) = \sinh xf′(x)=cosh⁡xf'(x) = \cosh x
f(x)=cosh⁡xf(x) = \cosh xf′(x)=sinh⁡xf'(x) = \sinh x
f(x)=tanh⁡−1xf(x) = \tanh^{-1} xf′(x)=11−x2f'(x) = \frac{1}{1 - x^2}
f(x)=sinh⁡−1xf(x) = \sinh^{-1} xf′(x)=11+x2f'(x) = \frac{1}{\sqrt{1 + x^2}}
f(x)=cosh⁡−1xf(x) = \cosh^{-1} xf′(x)=1x2−1f'(x) = \frac{1}{\sqrt{x^2 - 1}}

Integrales

IntegralResultado
∫dx\int dxx+Cx + C
∫xndx\int x^n dxxn+1n+1+C\frac{x^{n+1}}{n+1} + C
∫1xdx\int \frac{1}{x} dxln⁡∣x∣+C\ln |x| + C
∫exdx\int e^x dxex+Ce^x + C
∫sin⁡xdx\int \sin x dx−cos⁡x+C-\cos x + C
∫cos⁡xdx\int \cos x dxsin⁡x+C\sin x + C
∫axdx\int a^x dxaxln⁡a+C\frac{a^x}{\ln a} + C
∫sinh⁡xdx\int \sinh x dxcosh⁡x+C\cosh x + C
∫cosh⁡xdx\int \cosh x dxsinh⁡x+C\sinh x + C
∫sec⁡2xdx\int \sec^2 x dxtan⁡x+C\tan x + C
∫csc⁡2xdx\int \csc^2 x dx−cot⁡x+C-\cot x + C
∫sec⁡xtan⁡xdx\int \sec x \tan x dxsec⁡x+C\sec x + C
∫csc⁡xcot⁡xdx\int \csc x \cot x dx−csc⁡x+C-\csc x + C
∫11+x2dx\int \frac{1}{1+x^2} dxtan⁡−1x+C\tan^{-1} x + C
∫11−x2dx\int \frac{1}{\sqrt{1-x^2}} dxsin⁡−1x+C\sin^{-1} x + C
∫−11−x2dx\int \frac{-1}{\sqrt{1-x^2}} dxcos⁡−1x+C\cos^{-1} x + C
∫11−x2dx\int \frac{1}{1-x^2} dxtanh⁡−1x+C\tanh^{-1} x + C
∫11+x2dx\int \frac{1}{\sqrt{1+x^2}} dxsinh⁡−1x+C\sinh^{-1} x + C
∫1x2−1dx\int \frac{1}{\sqrt{x^2-1}} dxcosh⁡−1x+C\cosh^{-1} x + C

Hecho por Pablo Eguilaz 2025