Calculus Formulas

Cheatsheet Content

### Differentiation Rules - **Constant Rule:** $\frac{d}{dx}(c) = 0$ - **Power Rule:** $\frac{d}{dx}(x^n) = nx^{n-1}$ - **Constant Multiple Rule:** $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$ - **Sum/Difference Rule:** $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$ - **Product Rule:** $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$ - **Quotient Rule:** $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ - **Chain Rule:** $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ ### Derivatives of Polynomial and Rational Functions - $\frac{d}{dx}(x) = 1$ - $\frac{d}{dx}(x^n) = nx^{n-1}$ - $\frac{d}{dx}(c) = 0$ ### Derivatives of Trigonometric Functions - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(\tan x) = \sec^2 x$ - $\frac{d}{dx}(\cot x) = -\csc^2 x$ - $\frac{d}{dx}(\sec x) = \sec x \tan x$ - $\frac{d}{dx}(\csc x) = -\csc x \cot x$ ### Derivatives of Inverse Trigonometric Functions - $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$ - $\frac{d}{dx}(\text{arccot } x) = -\frac{1}{1+x^2}$ - $\frac{d}{dx}(\text{arcsec } x) = \frac{1}{|x|\sqrt{x^2-1}}$ - $\frac{d}{dx}(\text{arccsc } x) = -\frac{1}{|x|\sqrt{x^2-1}}$ ### Derivatives of Exponential and Logarithmic Functions - $\frac{d}{dx}(e^x) = e^x$ - $\frac{d}{dx}(a^x) = a^x \ln a$ - $\frac{d}{dx}(\ln x) = \frac{1}{x}$ - $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$ ### Integration Rules - **Constant Rule:** $\int c \, dx = cx + C$ - **Power Rule:** $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ - **Constant Multiple Rule:** $\int cf(x) \, dx = c\int f(x) \, dx$ - **Sum/Difference Rule:** $\int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx$ - **Substitution Rule:** $\int f(g(x))g'(x) \, dx = \int f(u) \, du$ (where $u=g(x)$) - **Integration by Parts:** $\int u \, dv = uv - \int v \, du$ ### Integrals of Polynomial and Rational Functions - $\int 1 \, dx = x + C$ - $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ - $\int \frac{1}{x} \, dx = \ln|x| + C$ ### Integrals of Trigonometric Functions - $\int \sin x \, dx = -\cos x + C$ - $\int \cos x \, dx = \sin x + C$ - $\int \tan x \, dx = -\ln|\cos x| + C = \ln|\sec x| + C$ - $\int \cot x \, dx = \ln|\sin x| + C$ - $\int \sec x \, dx = \ln|\sec x + \tan x| + C$ - $\int \csc x \, dx = -\ln|\csc x + \cot x| + C$ - $\int \sec^2 x \, dx = \tan x + C$ - $\int \csc^2 x \, dx = -\cot x + C$ - $\int \sec x \tan x \, dx = \sec x + C$ - $\int \csc x \cot x \, dx = -\csc x + C$ ### Integrals Leading to Inverse Trigonometric Functions - $\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{x\sqrt{x^2-a^2}} \, dx = \frac{1}{a}\text{arcsec}\left|\frac{x}{a}\right| + C$ ### Integrals of Exponential and Logarithmic Functions - $\int e^x \, dx = e^x + C$ - $\int a^x \, dx = \frac{a^x}{\ln a} + C$ - $\int \ln x \, dx = x\ln x - x + C$