Problem 56 - Thomas Calculus in SI Units

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: Understanding the Problem

We are given a function, \( y = 3^{-x} \), and asked to find the derivative \( \frac{dy}{dx} \). This involves using the rules of differentiation for exponential functions where the base is a constant.

Step 2: Applying the Exponential Differentiation Rule

Recall the differentiation rule for an exponential function: if \( y = a^{u(x)} \), then \( \frac{dy}{dx} = a^{u(x)} \ln(a) \cdot \frac{du}{dx} \). In our problem, \( a = 3 \) and \( u(x) = -x \).

Step 3: Calculating the Derivative of \( u(x) \)

We need to find \( \frac{du}{dx} \), where \( u(x) = -x \). The derivative of \( -x \) with respect to \( x \) is \( -1 \).

Step 4: Substituting into the Differentiation Rule

Substituting \( a = 3 \), \( u(x) = -x \), and \( \frac{du}{dx} = -1 \) into the exponential differentiation formula, we have: \( \frac{dy}{dx} = 3^{-x} \ln(3) \cdot (-1) \).

Step 5: Simplifying the Expression

Simplify \( \frac{dy}{dx} = 3^{-x} \ln(3) \cdot (-1) \) to get \( \frac{dy}{dx} = -3^{-x} \ln(3) \). This is the final expression for the derivative.

Key Concepts

Exponential Functions

Exponential functions are an essential component of calculus and mathematical analysis. These functions have the general form of \( f(x) = a^{x} \), where the base \( a \) is a positive constant. The variable, usually denoted as \( x \), is found in the exponent position, which gives the function its unique exponential growth or decay characteristics.
Exponential growth occurs when the base \( a \) is greater than 1. This means that as \( x \) increases, \( f(x) \) grows larger rapidly. Conversely, when \( 0 < a < 1 \), the function models exponential decay, where the function values decrease and get closer to zero as \( x \) becomes larger. Notable attributes of exponential functions include:

• Their range is always positive: \( (0, \infty) \).
• They pass through the point \( (0, 1) \) because \( a^{0} = 1 \).
• They have continuous growth or decay rates.

Understanding these characteristics will help you in calculus when learning about their derivatives and integrals.

Derivative Rules

Derivative rules are the guidelines we use to find the rate of change of a function with respect to a variable. When dealing with exponential functions like \( y = a^{u(x)} \), a special rule applies. The rule is: the derivative of \( y = a^{u(x)} \) with respect to \( x \) is \( \frac{dy}{dx} = a^{u(x)} \ln(a) \cdot \frac{du}{dx} \).
This rule comes from the fundamental concept that the derivative measures how a function changes as its inputs change. In this case, we multiply the original function by the natural logarithm of its base and then by the derivative of the exponent function \( u(x) \).
Keep the following steps in mind when differentiating exponential functions:

• Identify the base \( a \) and the exponent function \( u(x) \).
• Differentiate \( u(x) \) to find \( \frac{du}{dx} \).
• Apply the exponential differentiation formula.

Using these rules ensures precise calculation of derivatives, which is crucial in various applications like growth models and decay processes.

Calculus Problem Solving

Calculus problem solving involves applying various differentiation rules to actual problems. Let's take our example of finding the derivative of \( y = 3^{-x} \). First, identify that this is an exponential function where the base \( a \) is 3 and the exponent function \( u(x) = -x \).
The problem-solving approach includes:

• Recognizing the type of function and selecting the appropriate differentiation rule.
• Computing the derivative of the exponent, which in this case is \( \frac{du}{dx} = -1 \).
• Substituting these values into the rule to determine \( \frac{dy}{dx} \).

After substitution, you get \( \frac{dy}{dx} = 3^{-x} \ln(3) \cdot (-1) \). Simplifying this expression gives you the final answer: \( \frac{dy}{dx} = -3^{-x} \ln(3) \).
Solving calculus problems often depends on understanding function types and applying the fitting rules to find derivatives, ensuring accuracy in mathematical modeling and analysis.