Calculus Formulas (Derivatives, Integrals, FTC)

$$\text{Definition of derivative: } f'(x) =$$

$$\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$$

$$\text{Alternate definition: } f'(a) =$$

$$\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$$

$$\dfrac{d}{dx}[c] =$$

$$0$$

$\dfrac{d}{dx}\left[\dfrac{f}{g}\right] =$

$\dfrac{gf'-fg'}{g^2}$

$\dfrac{d}{dx}[fg] =$

$fg'+f'g$

$\dfrac{d}{dx}[e^x] =$

$e^x$

$\dfrac{d}{dx}[x] =$

$1$

$\dfrac{d}{dx}[u+v] =$

$u'+v'$

$\dfrac{d}{dx}[u^n] =$

$nu^{n-1}\,u'$

$\dfrac{d}{dx}[\cos u] =$

$-\sin u\,u'$

$\dfrac{d}{dx}[\sin u] =$

$\cos u\,u'$

$\dfrac{d}{dx}[\sec u] =$

$\sec u\tan u\,u'$

$\dfrac{d}{dx}[\tan u] =$

$\sec^2 u\,u'$

$\dfrac{d}{dx}[\cot u] =$

$-\csc^2 u\,u'$

$\dfrac{d}{dx}[\csc u] =$

$-\csc u\cot u\,u'$

$\dfrac{d}{dx}[\ln u] =$

$\dfrac{u'}{u}$

$\dfrac{d}{dx}[\log_a u] =$

$\dfrac{u'}{(\ln a)\,u}$

$\dfrac{d}{dx}[a^u] =$

$(\ln a)\,a^u\,u'$

$\dfrac{d}{dx}[\arcsin u] =$

$\dfrac{u'}{\sqrt{1-u^2}}$

$\dfrac{d}{dx}[\arctan u] =$

$\dfrac{u'}{1+u^2}$

$\dfrac{d}{dx}[\operatorname{arcsec} u] =$

$\dfrac{u'}{|u|\sqrt{u^2-1}}$

$\int k\,f(x)\,dx =$

$k\int f(x)\,dx$

$\int [f(x)\pm g(x)]\,dx =$

$\int f(x)\,dx\pm\int g(x)\,dx$

$\int dx =$

$x+C$

$\int u^n\,du =$

$\dfrac{u^{n+1}}{n+1}+C$

$\int \dfrac{du}{u} =$

$\ln|u|+C$

$\int e^u\,du =$

$e^u+C$

$\int a^u\,du =$

$\left(\dfrac{1}{\ln a}\right)a^u+C$

$\int \sin u\,du =$

$-\cos u+C$

$\int \cos u\,du =$

$\sin u+C$

$\int \tan u\,du =$

$-\ln|\cos u|+C$

$\int \cot u\,du =$

$\ln|\sin u|+C$

$\int \sec u\,du =$

$\ln|\sec u+\tan u|+C$

$\int \csc u\,du =$

$-\ln|\csc u+\cot u|+C$

$\int \sec^2 u\,du =$

$\tan u+C$

$\int \csc^2 u\,du =$

$-\cot u+C$

$\int \sec u\tan u\,du =$

$\sec u+C$

$\int \csc u\cot u\,du =$

$-\csc u+C$

$\int \dfrac{du}{\sqrt{a^2-u^2}} =$

$\arcsin\dfrac{u}{a}+C$

$\int \dfrac{du}{a^2+u^2} =$

$\dfrac{1}{a}\arctan\dfrac{u}{a}+C$

$\int \dfrac{du}{u\sqrt{u^2-a^2}} =$

$\dfrac{1}{a}\operatorname{arcsec}\dfrac{|u|}{a}+C$

$$\int \ln x\,dx =$$

$$x\ln x-x+C$$

$$\text{FTC: } \int_a^b f'(x)\,dx =$$

$$f(b)-f(a)$$

$$\text{FTC: if } F'(x)=f(x)\text{, then } \int_a^b f(x)\,dx =$$

$$F(b)-F(a)$$