# [[Exponential Function and Its Integral]] (Mathematics > Calculus) ## Definition The exponential function $e^x$ is unique because its derivative and integral are both the same as the function itself. Specifically, the derivative of $e^x$ with respect to $x$ is: $ \frac{d}{dx}(e^x) = e^x $ Similarly, the integral of $e^x$ is: $ \int e^x \, dx = e^x + C $ ## Key Concepts - The exponential function $e^x$ remains unchanged when differentiated or integrated. - The constant $C$ is added to the result of integration to account for indefinite integration. - The base of the natural exponential function, $e$, is approximately $2.718$. ## Important Properties 1. $e^x$ is its own derivative and integral: $\frac{d}{dx}(e^x) = e^x$ and $\int e^x dx = e^x + C$. 2. The exponential function grows rapidly as $x$ increases. 3. For negative values of $x$, $e^x$ approaches $0$ but never reaches it. ## Essential Formulas - Derivative: $ \frac{d}{dx}(e^x) = e^x $ - Integral: $ \int e^x \, dx = e^x + C $ ## Core Examples 1. Compute the derivative of $e^{3x}$: $ \frac{d}{dx} e^{3x} = 3e^{3x} \quad \text{(by the chain rule)} $ 2. Evaluate the integral of $5e^x$: $ \int 5e^x \, dx = 5e^x + C $ ## Related Theorems/Rules - [[Chain Rule]]: Used when differentiating composite functions like $e^{kx}$. - [[Fundamental Theorem of Calculus]]: Relates the processes of differentiation and integration. ## Common Pitfalls - Forgetting to include the constant $C$ when performing indefinite integration. - Confusing the derivative and integral of $e^x$ with other exponential functions that have different bases. ## Related Topics - [[Derivative]] - [[Power Rule]] - [[Chain Rule]] - [[Derivative of the Exponential Function]] - [[Derivative of e_x Using the Limit Definition]] - [[Antiderivatives]] - [[Indefinite Integrals]] - [[Power Rule for Integration]] - [[Natural Logarithm and Its Derivative]] - [[Logarithmic Functions]] - [[Growth and Decay Models]] ## Quick Review Questions 1. What is the derivative of $e^{-x}$? 2. What is the integral of $7e^x$? *** ![[Exponential_Function_and_Its_Integral_visualization]]