Problem 33 - Calculus Single Variable

Question

Calculate the derivative of the given xpression with respect to \(x\). $$ 3^{2 x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the General Form

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

Step 2: Differentiate Using the Chain Rule

To differentiate \(3^{2x}\), we use the chain rule. The derivative of \(a^{u(x)}\) with respect to \(x\) is \(a^{u(x)} \ln(a) \cdot \frac{du}{dx}\).

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

Since \(u(x) = 2x\), the derivative \(\frac{du}{dx}\) is \(2\).

Step 4: Apply the Derivatives

Substituting back into the chain rule formula, we have:\[\frac{d}{dx} 3^{2x} = 3^{2x} \ln(3) \cdot 2.\]

Step 5: Simplify the Expression

After performing the multiplication, the derivative of \(3^{2x}\) is:\[2 \cdot 3^{2x} \ln(3).\]

Key Concepts

Exponential Functions

Exponential functions are an intriguing part of calculus and mathematics in general. These functions are written in the form \(a^{x}\), where \(a\) is a constant base and \(x\) is the exponent or power. In our example, the function given is \(3^{2x}\). This means the base \(3\) is raised to the power of \(2x\).

Exponential functions grow or decay at rates proportional to their current values. They are widely used in various fields such as biology, finance, and physics.

• In biology, they describe population growth.
• In finance, they are used for compound interest calculations.
• In physics, they appear in phenomena like radioactive decay.

When dealing with exponential functions in calculus, particularly regarding differentiation and integration, it's crucial to understand how they behave differently from polynomial functions. For instance, while the derivative of \(x^n\) is \(nx^{n-1}\), exponential functions have their own unique differentiation rules, often involving natural logarithms for efficiency and accuracy.

Chain Rule

The chain rule is an essential differentiation technique used when dealing with a composition of functions. It allows us to calculate the derivative of composite functions efficiently.

Imagine we have a function \(y = f(g(x))\). The chain rule states that the derivative of \(y\) with respect to \(x\) is given by: \[\frac{dy}{dx} = f'(g(x)) \, g'(x)\] In this rule:

• \(f'(g(x))\) is the derivative of the outer function with respect to its argument.
• \(g'(x)\) is the derivative of the inner function.

Applying this to our exercise with \(3^{2x}\), we identify \(u(x) = 2x\) as the inner function \(g(x)\) and \(3^{u(x)}\) as \(f(u)\). The chain rule guides us to take the derivative of the outer function and then multiply it by the derivative of the inner function. This method ensures we correctly handle the function's composition without errors.

Differentiation Techniques

Mastering differentiation techniques is key to solving problems involving functions, as it allows us to find rates of change and slopes of curves. In the context of the given exercise, we used a combination of the chain rule and the knowledge of exponential functions' derivative properties.

Here is a brief summary of important differentiation techniques related to exponential functions:

• Basic Rule for Exponential Functions: The derivative of \(a^{x}\) is \(a^{x} \ln(a)\), where \(\ln(a)\) is the natural logarithm of the base \(a\). This rule stems from the unique way exponential functions grow.
• Using the Chain Rule: When the exponent is more complex, like \(2x\) in \(3^{2x}\), apply the chain rule to differentiate correctly. This involves finding the derivative of the exponential \(a^{u(x)}\), followed by multiplying with \(\frac{du}{dx}\).

To simplify the derivative calculation, remember to:

• Determine the expression’s structure, as seen with \(3^{2x}\).
• Identify and differentiate both the inner and outer functions using the appropriate rules.
• Always check your work by simplifying the result, like obtaining \(2 \cdot 3^{2x} \, \ln(3)\) from the original expression.

These techniques not only help in mathematical exercises but are also applied across various scientific computations.