Problem 68 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{-x}$$

Step-by-Step Solution

Answer

The derivative is \(-3^{-x} \ln(3)\).

Step 1: Identify the Problem

We need to find the derivative of the given function, which is expressed as an exponential function: \( y = 3^{-x} \). The goal is to determine \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \).

Step 2: Apply the Exponential Derivative Rule

To find the derivative of an expression of the form \( a^{f(x)} \), use the rule: \( \frac{d}{dx}[a^{f(x)}] = a^{f(x)} \ln(a) \cdot f'(x) \), where \( a \) is a constant. Here, \( a = 3 \) and \( f(x) = -x \).

Step 3: Calculate the Derivative of the Exponent

Find the derivative of \( f(x) = -x \) with respect to \( x \). The derivative, \( f'(x) \), is simply \( -1 \) since the derivative of \( x \) is 1 and we have a negative sign.

Step 4: Substitute into the Derivative Formula

Substitute \( f(x) = -x \), \( f'(x) = -1 \), and \( a = 3 \) into the exponential derivative formula: \[ \frac{dy}{dx} = 3^{-x} \ln(3) \cdot (-1) \].

Step 5: Simplify the Expression

Simplify the expression for the derivative: \[ \frac{dy}{dx} = -3^{-x} \ln(3) \]. This is the simplified expression for the derivative of the given function.

Key Concepts

Exponential Function

An exponential function is a type of mathematical expression where the variable is the exponent. This function takes the form of \( y = a^x \), where \( a \) is a positive constant, known as the base, and \( x \) is the exponent. Exponential functions are widely used in different fields such as finance, biology, and physics because they describe exponential growth or decay.

• For growth, the base \( a \) is greater than 1, indicating that as \( x \) increases, so does \( y \).
• For decay, the base \( a \) is between 0 and 1, meaning that as \( x \) increases, \( y \) decreases.

In our original exercise, the function \( y = 3^{-x} \) is an example of exponential decay because the exponent \(-x\) causes the value of \( y \) to decrease as \( x \) increases.

Exponential Derivative Rule

The exponential derivative rule is a handy tool for finding the derivative of an exponential function. This rule states that if you have a function in the form of \( a^{f(x)} \), its derivative is given by:

• \( \frac{d}{dx}[a^{f(x)}] = a^{f(x)} \ln(a) \cdot f'(x) \)

Here is a breakdown of the components:

• \( a^{f(x)} \) stays in the expression, meaning the exponential part remains the same.
• \( \ln(a) \) is the natural logarithm of the base, which is constant for a specific \( a \).
• \( f'(x) \) is the derivative of the exponent \( f(x) \), detailing how \( x \) changes.

In the given problem, since \( f(x) = -x \), we use this rule by substituting the appropriate values for \( f(x) \) and its derivative \( f'(x) \).
This allows us to compute \( \frac{dy}{dx} \) with ease.

Derivative of Exponential Functions

Finding the derivative of an exponential function, particularly one with a negative exponent as given in \( y = 3^{-x} \), involves applying the exponential derivative rule correctly.

• The function \( y = 3^{-x} \) maps onto the rule as \( a^{f(x)} = 3^{-x} \).
• The derivative of \( f(x) = -x \) is \( f'(x) = -1 \).
• Substituting these values back into the derivative rule results in:
  \( \frac{dy}{dx} = 3^{-x} \ln(3) \cdot (-1) \).

By simplifying this, we see that the negative sign from \( f'(x) = -1 \) factors into the result, giving us the final expression:
\( \frac{dy}{dx} = -3^{-x} \ln(3) \).
This simplified form provides a clear understanding of how the rate of change in \( y \) corresponds to changes in \( x \), while taking into account the base and its natural logarithm.