Derivative of e^2x: Answer, Chain Rule, Mistakes

How to differentiate e^(2x)

The outer function is the exponential, which is its own derivative, and the inner function is $2x$.

Why this function solves a differential equation

The derivative is exactly $2$ times the original function, so $y = e^{2x}$ satisfies $\frac{dy}{dx} = 2y$. That is the defining property of exponential growth, and it is why exponentials solve these equations.

A negative constant gives decay instead: $\frac{d}{dx}e^{-2x} = -2e^{-2x}$, and the function falls toward zero.

Where the derivative of e^(2x) shows up on the AP exam

Topic 3.1 for the chain rule, and Unit 7 for growth and decay, on both exams. Reversed it gives $\int e^{2x}\,dx = \frac{e^{2x}}{2} + C$, where the same $2$ divides rather than multiplies.

Common mistakes with the derivative of e^(2x)

• Answering $e^{2x}$ and forgetting the chain rule factor.
• Answering $2xe^{2x-1}$, applying the power rule to something that is not a power.
• Confusing it with $\left(e^{x}\right)^{2}$, which is the same function, and with $e^{x^{2}}$, which is not: that one differentiates to $2xe^{x^{2}}$.