In calculus, the exponential function $e^x$ is unique because its derivative is exactly itself ($e^x$). However, when the exponent is modified to something like $2x$, you must apply the Chain Rule to find the correct derivative.
The number $e$ is Euler's Number, an irrational mathematical constant approximately equal to 2.71828.
The integration (anti-derivative) of $e^{2x}$ does the exact opposite. Instead of multiplying by 2, you divide: $\int e^{2x} dx = \frac{e^{2x}}{2} + C$.
The derivative of $e^{2x}$ with respect to $x$ is $2e^{2x}$.
Mathematically: $\frac{d}{dx}(e^{2x}) = 2e^{2x}$
The Chain Rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.
Step 1: Differentiate the outer function The derivative of $e^u$ is simply $e^u$. So, the first part is $e^{2x}$.
Step 2: Differentiate the inner function The derivative of $2x$ with respect to $x$ is just the constant $2$. So, $\frac{du}{dx} = 2$.
Step 3: Multiply them together According to the chain rule, we multiply the result of Step 1 by the result of Step 2. $= e^{2x} \cdot 2$ $= 2e^{2x}$
You can memorize a shortcut rule for any exponential function with a constant multiplier in the power: $\frac{d}{dx}(e^{kx}) = k \cdot e^{kx}$ (Where $k$ is any constant number. So if it was $e^{5x}$, the answer is simply $5e^{5x}$).
Using the same chain rule, the derivative of the inner function (-x) is -1. So, the final derivative is -1 · e^(-x) or -e^(-x).
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