Calculus Basics

Differentiation

At all points $x$ where the functions and the derivatives are defined,

\frac{d}{dx}(x^n) = nx^{n-1} \quad \frac{d}{dx}\sin(x) = \cos(x) \quad \frac{d}{dx}\cos(x) = -\sin(x) \quad \frac{d}{dx}\tan(x) = \sec^2(x)

\frac{d}{dx}\sec(x) = \sec(x)\tan(x) \quad \frac{d}{dx}\csc(x) = -\csc(x)\cot(x) \quad \frac{d}{dx}\cot(x) = -\csc^2(x)

\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}} \quad \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2} \quad \frac{d}{dx}\text{arcsec}(x) = \frac{1}{|x|\sqrt{1-x^2}}

\frac{d}{dx}(e^x) = e^x \quad \frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x) \quad \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + g'(x)f(x)

\frac{d}{dx}(\ln(x)) = \frac{1}{x} \quad \frac{d}{dx}f(g(x)) = f'(g(x))g'(x) \quad \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

\frac{d}{dx}(f(x)^{g(x)}) = f(x)^{g(x)} \left[g'(x)\ln(f(x)) + g(x) \cdot \frac{f'(x)}{f(x)}\right] \quad \frac{d}{dx}(x^x) = x^x(\ln(x) + 1)

Integration

Disregarding the $+C$ on all the integrals,

\int x^n dx = \frac{x^{n+1}}{n+1}, n \neq -1 \quad \int \sin(x) dx = -\cos(x) \quad \int \cos(x) dx = \sin(x) \quad \int \tan(x) dx = -\ln|\cos(x)|

\int \sec(x) dx = \ln|\sec(x) + \tan(x)| \quad \int \csc(x) dx = \ln|\csc(x) - \cot(x)| \quad \int \cot(x) dx = \ln|\sin(x)|

\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x) \quad \int \frac{1}{1+x^2} dx = \arctan(x) \quad \int \frac{1}{|x|\sqrt{1-x^2}} dx = \text{arcsec}(x)

\int e^x dx = e^x \quad \int \frac{1}{x} dx = \ln|x| \quad \int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx

\int u(x)v'(x) dx = u(x)v(x) - \int v(x)u'(x) dx \quad \int f'(g(x))g'(x) dx = f(g(x))

Taylor Series

Select some point $x = x_0$ . If $x_0 = 0$ , we have the Maclaurin series. Generally, $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n$ . Common Maclaurin series expansions:

e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots

\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots

\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots

Common Summation Formulae

\sum_{k=1}^n k = \frac{n(n+1)}{2} \quad \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} \quad \sum_{k=s}^\infty a \cdot r^k = a \cdot \frac{r^s}{1-r} \quad \sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}