Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(6^{2 x}\right) $$

Step-by-Step Solution

Answer

The derivative is \(2 \cdot \ln(6) \cdot 6^{2x}\).

Step 1: Recall the derivative rule for exponential functions

The derivative of an exponential function of the form \(a^{u(x)}\) with respect to \(x\) is given by \(a^{u(x)} \cdot u'(x) \cdot \ln(a)\), where \(a\) is a constant and \(u(x)\) is a function of \(x\).

Step 2: Identify the function components

In the function \(6^{2x}\), \(a = 6\) and \(u(x) = 2x\). This means our task is to differentiate with respect to \(x\).

Step 3: Differentiate the exponent

Differentiating \(u(x) = 2x\) with respect to \(x\), we get \(u'(x) = 2\).

Step 4: Apply the derivative rule

Substitute \(a = 6\), \(u(x) = 2x\), and \(u'(x) = 2\) into the rule. The derivative becomes: \[D_x(6^{2x}) = 6^{2x} \cdot 2 \cdot \ln(6)\].

Step 5: Simplify the result

The final simplified derivative is:\[2 \cdot \ln(6) \cdot 6^{2x}\].

Key Concepts

Exponential Functions

Exponential functions are a crucial concept in calculus and many other fields of mathematics. At their core, these functions involve a base number raised to a variable exponent. For example, the function \( 6^{2x} \) is an exponential function where 6 is the base and \( 2x \) is the exponent. Unlike linear or polynomial functions, the variable is in the exponent, making these functions grow or decay at exponential rates.
These functions are characterized by rapid increases or decreases in their values. They are ubiquitous in real-life scenarios such as population growth, radioactive decay, and interest calculations in banking.

• The base can be any positive number except 1, but common bases include 2, \( e \) (approximately 2.718), and 10.
• When differentiating, or taking the derivative of an exponential function, the chain rule or the power of \( e \) is often involved depending on the function's form.

By understanding how these functions work, you can make sense of their unique properties and apply them effectively in solving real-world problems.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to a variable. In simpler terms, it tells us how fast or slow a function is changing at any given point.
To differentiate exponential functions, we use specific rules that help us find the derivative efficiently. For a function of the form \( a^{u(x)} \):

• The differentiation rule states that the derivative is \( a^{u(x)} \cdot u'(x) \cdot \ln(a) \), where \( u'(x) \) is the derivative of the exponent function \( u(x) \), and \( \ln(a) \) is the natural logarithm of the base.
• This rule requires understanding how to handle both the power \( u(x) \) and the natural logarithm.

For instance, in the function \( 6^{2x} \), by identifying \( a = 6 \), \( u(x) = 2x \), and \( u'(x) = 2 \), the differentiation becomes straightforward. It is a perfect example of applying rules specific to exponential functions and logarithmic differentiation.

Mathematical Simplification

Mathematical simplification involves reducing expressions to their simplest form. This process helps in making complex problems more manageable and the results more interpretable. In calculus, especially during differentiation, simplification is often the final step after computing a derivative.
Consider the derivative found for \( 6^{2x} \). Initially, it was derived as \( 6^{2x} \cdot 2 \cdot \ln(6) \). Simplifying this means arranging or rewriting it to highlight its intrinsic properties or for ease of use.

• Simplification can involve combining like terms, factoring, and canceling out unnecessary elements—essentially refining the expression to its most elegant form.
• This task highlights understanding exponential terms and logarithmic operations within the differentiation context. The outcome, \( 2 \cdot \ln(6) \cdot 6^{2x} \), is a neat expression that retains all crucial information.

Proper simplification aids in recognizing patterns or contributes to clearer insight and communication of the solution. It is a valuable skill in mathematics, making results more approachable and inclusive for any audience.