What is the Derivative of $e^{2x}$?

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 $f'(g(x)) \cdot g'(x)$.

• Let our outer function be the exponential: $f(u) = e^u$
• Let our inner function be the exponent: $u = 2x$

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 $5e^{5x}$).