\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\]

Introduction:

Finding the derivative of \( 2^x \) helps us understand how the function changes as \( x \) changes. Unlike derivatives of polynomials or basic trigonometric functions, differentiating an exponential function involves the natural logarithm. Let’s walk through the steps.

Step-by-Step Solution:

To differentiate \( 2^x \), it helps to rewrite it using the base \( e \) by using the identity:

\[\begin{align*} 2^x = e^{x \ln(2)} \end{align*}\]

This is valid because \( e^{x \ln(2)} \) is equivalent to \( 2^x \), since \( e^{\ln(2)} = 2 \). Now, we differentiate \( e^{x \ln(2)} \) with respect to \( x \):

\[\begin{align*} \frac{d}{dx} \left( e^{x \ln(2)} \right) = e^{x \ln(2)} \cdot \ln(2) \end{align*}\]

In this step, we used the chain rule on $e^x$, which states that the derivative of \( e^{f(x)} \) is \( e^{f(x)} \cdot f'(x) \). Here, \( f(x) = x \ln(2) \), and its derivative \( f'(x) = \ln(2) \). Applying this to \( 2^x \) and simplifying $e^{x\ln(2)}$, we get that the derivative is:

\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\]

Why This Works:

When differentiating exponential functions with bases other than \( e \), the natural logarithm of the base appears as a multiplier. Here, the base is 2, so \( \ln(2) \) shows up in the answer. This result generalizes to any exponential function \( a^x \), where \( \frac{d}{dx} \left( a^x \right) = a^x \ln(a) \). This formula is essential in calculus and helps to differentiate and understand the behavior of exponential functions.