In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x

Verified step by step guidance

1. Recognize that the function is an exponential function of the form \(y = a^x\) where \(a = 2\) is a constant base and \(x\) is the variable.
2. Recall the general formula for the derivative of an exponential function with a constant base: \(\frac{d}{dx} a^x = a^x \ln(a)\).
3. Apply this formula to the given function by substituting \(a = 2\), so the derivative becomes \(\frac{dy}{dx} = 2^x \ln(2)\).
4. Understand that \(\ln(2)\) is the natural logarithm of 2, which is a constant multiplier in the derivative.
5. Write the final expression for the derivative as \(\frac{dy}{dx} = 2^x \ln(2)\), which completes the differentiation process.

Key Concepts

Derivative of Exponential Functions

The derivative of an exponential function with a constant base, such as y = a^x, involves the natural logarithm of the base. Specifically, the derivative is dy/dx = a^x * ln(a), where ln(a) is the natural log of the base a. This rule helps differentiate functions like y = 2^x.

Natural Logarithm (ln)

The natural logarithm, denoted ln(x), is the inverse of the exponential function with base e. It is essential in differentiation of exponential functions with bases other than e, as it appears in the derivative formula to account for the base's growth rate.

Basic Differentiation Rules

Understanding basic differentiation rules, such as the power rule and constant multiple rule, is crucial. These rules allow you to differentiate functions systematically and combine with the exponential derivative formula to find the derivative of more complex expressions.