Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(3^{2 x^{2}-3 x}\right) $$

Step-by-Step Solution

Answer

The derivative is \( 3^{2x^2-3x} \cdot \ln(3) \cdot (4x - 3) \).

Step 1: Identify the Function Type

The function given is an exponential function of the form \( a^{u(x)} \), where \( a = 3 \) and \( u(x) = 2x^2 - 3x \).

Step 2: Apply the Exponential Derivative Rule

The derivative of \( a^{u(x)} \) with respect to \( x \) is \( a^{u(x)} \cdot \ln(a) \cdot u'(x) \). We will apply this rule to find the derivative.

Step 3: Differentiate \( u(x) \)

Find the derivative of \( u(x) = 2x^2 - 3x \) with respect to \( x \). \[ u'(x) = \frac{d}{dx}(2x^2) - \frac{d}{dx}(3x) = 4x - 3 \]

Step 4: Substitute Into the Exponential Derivative Rule

Substitute \( u(x) \), \( u'(x) \), and \( a = 3 \) into the derivative formula:\[ D_{x}(3^{2x^2-3x}) = 3^{2x^2-3x} \cdot \ln(3) \cdot (4x - 3) \].

Step 5: Simplify the Expression

The derivative is \( 3^{2x^2-3x} \cdot \ln(3) \cdot (4x - 3) \). No further simplification is needed for this expression.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In more specific terms, functions of this type can be written as \( a^{u(x)} \), where \( a \) is a constant and \( u(x) \) is a function of \( x \).
Exponential functions are vital in various fields, such as sciences, economics, and statistics, due to their unique property of constant relative growth. For example, compound interest calculations in finance frequently use exponential functions.
In our exercise, the function \( 3^{2x^2-3x} \) represents an exponential function where the base \( a = 3 \) and the exponent is the polynomial \( u(x) = 2x^2 - 3x \). Understanding this setup is the first step in determining the derivative.

• The base of exponential functions is fixed, while only the power varies depending on the variable in question.
• These functions can increase or decrease exponentially based on the base and the exponent's sign.

Differentiation Rules

Differentiation is a fundamental concept in calculus, allowing us to find the rate at which a function changes. To differentiate exponential functions, we apply specific rules that consider both the base and the exponent.
For our problem, the rule we use for differentiation of \( a^{u(x)} \) is \( a^{u(x)} \cdot \ln(a) \cdot u'(x) \). Here's why this rule works:

• \( \ln(a) \) comes from the derivative of the exponential function itself.


• \( u'(x) \) is the derivative of the exponent, capturing any rate of change happening due to \( x \).

Applying this rule requires us to know how to differentiate polynomial functions, like \( u(x) = 2x^2 - 3x \), which we calculated as \( 4x - 3 \). It's essential to first find \( u'(x) \) before using it in the exponential derivative rule.

• The differentiation formula helps break down complex functions into simpler parts.


• It is constructive practice to ensure each component, like the derivative of \( u(x) \), is correctly determined before plugging it back into the formula.

Step-by-Step Solution

Breaking down the solution into manageable steps makes the problem easier to understand and solve. Here’s a step-by-step outline on how to tackle similar problems:
1. **Identify the Function Type:** Recognize that the function is exponential based on its structure. Knowing this helps determine which differentiation rules to apply.
2. **Apply the Exponential Derivative Rule:** Once identified, use the derivative rule for exponential functions, \( a^{u(x)} \cdot \ln(a) \cdot u'(x) \).
3. **Differentiate \( u(x) \):** Find the derivative of the exponent function separately. In our case, that was \( 4x - 3 \).
4. **Substitute and Simplify:** Substitute all known values and derivatives into the exponential differentiation formula and simplify the expression if necessary to arrive at the solution.
The final derivative for this exercise ends up being \( 3^{2x^2-3x} \cdot \ln(3) \cdot (4x - 3) \). Each of these steps ensures that we systematically approach complex problems, turning them into simpler parts that are easier to solve individually.

• Step-by-step methods prevent errors by keeping calculations clear and organized.
• Ensuring understanding of each step aids in effectively solving various calculus problems, enhancing skills in differentiation.