\section*{Calculus Formula Sheet: Essential Equations & Applications}

Master calculus with our comprehensive formula sheet covering derivatives, integrals, and their applications. This guide is essential for students studying calculus, engineering, physics, and related fields.

\subsection*{Derivatives}

\subsubsection*{Basic Derivative Rules} \begin{itemize} \item Power Rule: $\frac{d}{dx}[x^n] = n x^{n-1}$ \item Constant Rule: $\frac{d}{dx}[c] = 0$ \item Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$ \item Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ \item Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ \end{itemize}

\subsubsection*{Common Derivatives} \begin{itemize} \item $\frac{d}{dx}[\sin(x)] = \cos(x)$ \item $\frac{d}{dx}[\cos(x)] = -\sin(x)$ \item $\frac{d}{dx}[e^x] = e^x$ \item $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ \item $\frac{d}{dx}[\tan(x)] = \sec^2(x)$ \end{itemize}

\subsection*{Integrals}

\subsubsection*{Basic Integration Rules} \begin{itemize} \item Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) \item Constant Rule: $\int c dx = cx + C$ \item Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$ \end{itemize}

\subsubsection*{Common Integrals} \begin{itemize} \item $\int \sin(x) dx = -\cos(x) + C$ \item $\int \cos(x) dx = \sin(x) + C$ \item $\int e^x dx = e^x + C$ \item $\int \frac{1}{x} dx = \ln|x| + C$ \item $\int \sec^2(x) dx = \tan(x) + C$ \end{itemize}

\subsection*{Applications}

\subsubsection*{Area Under a Curve} The area between the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$ is: [A = \int_a^b f(x) dx]

\subsubsection*{Volume of Revolution} The volume generated by rotating $y = f(x)$ around the x-axis from $x = a$ to $x = b$ is: [V = \pi \int_a^b [f(x)]^2 dx]

\subsubsection*{Work Done by a Force} If $F(x)$ is a force function, the work done from $x = a$ to $x = b$ is: [W = \int_a^b F(x) dx]

\subsection*{Keywords} calculus formulas, derivatives, integrals, calculus cheat sheet, calculus equations

\subsection*{Last Updated} July 12, 2025, 05:38 PM +04