Find the derivative of f(x) = 2e^(3x).

Step-by-step solution

1. 1

   Separate the three ingredients. The function is a constant $2$ times a composite $e^{3x}$, whose outer function is $e^{u}$ and inner function is $u=3x$. Constants factor straight through differentiation, so the real work is the composite.

2. 2

   Differentiate the outer exponential. The exponential is its own derivative:

   $\frac{d}{du}e^{u}=e^{u}$

   so no new function appears — only a factor from the chain rule can change the answer.

3. 3

   Differentiate the inner function and apply the chain rule.

   $\frac{d}{dx}(3x)=3\ \Longrightarrow\ \frac{d}{dx}e^{3x}=3e^{3x}$

   Forgetting this factor of 3 — writing $e^{3x}$ as its own derivative — is the single most common mistake with exponentials.

4. 4

   Multiply by the constant.

   $\frac{d}{dx}\left(2e^{3x}\right)=2\cdot 3e^{3x}=6e^{3x}$

5. 5

   Generalise and check. The same reasoning gives $\dfrac{d}{dx}\left(Ce^{kx}\right)=kCe^{kx}$ for any constants $C,k$: the exponential shape survives and only the coefficient changes. Numerically at $x=0.4$ a central difference gives $19.9207$, and $6e^{1.2}=19.9207$ ✓.