Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

The given function is an exponential function: \( f(x) = 3^{2x^2 - 3x} \). We need to find the derivative of this function with respect to \( x \).

Step 2: Rewrite the Exponential Function using the Natural Exponent

Recall that any exponential function \( a^x \) can be rewritten using the natural number \( e \) as \( e^{x \ln(a)} \). Apply this to the given function: \( f(x) = e^{(2x^2 - 3x) \ln(3)} \).

Step 3: Differentiate using the Chain Rule

To differentiate \( f(x) = e^{u} \) where \( u = (2x^2 - 3x) \ln(3) \), we use the chain rule: \( \frac{d}{dx}[e^u] = e^u \cdot \frac{du}{dx} \).

Step 4: Find the Derivative of u with respect to x

The function \( u = (2x^2 - 3x) \ln(3) \) is a product. Differentiate \( u \) as \( \frac{du}{dx} = \ln(3) \cdot (4x - 3) \).

Step 5: Combine Results for the Derivative

Plug \( e^u = e^{(2x^2 - 3x) \ln(3)} = 3^{2x^2 - 3x} \) and \( \frac{du}{dx} = \ln(3) \cdot (4x - 3) \) into the chain rule result: \( D_x(3^{2x^2 - 3x}) = 3^{2x^2 - 3x} \cdot \ln(3) \cdot (4x - 3) \).

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus, especially when dealing with composite functions. Imagine function g(x) is embedded within another function f(u). To find the derivative of the entire composition, we use the chain rule. The chain rule formula is:

• \( \frac{d}{dx}\big[ f(g(x)) \big] = f'(g(x)) \cdot g'(x) \)

This means we differentiate the outer function, while keeping the inner function unchanged, and then multiply by the derivative of the inner function.

For example, if you have a situation like in the exercise where the function \( f(x) = e^u \) with \( u = (2x^2 - 3x) \ln(3) \), the chain rule suggests that we differentiate \( e^u \) treating \( u \) as constant, then multiply by \( \frac{du}{dx} \), the derivative of \( u \).

This is helpful when differentiating complex functions expressed in terms of simpler functions, like exponential functions in our task.

Exponential Functions

Exponential functions represent a key area of calculus due to their unique properties, particularly their rate of growth. A general form of such a function is \( a^x \), where \( a \) is a constant, and \( x \) is the variable exponent.

The exponential function is notable because its derivative has a specific property: the derivative of \( e^x \) is \( e^x \) itself. For other bases, such as \( 3^x \), the derivative involves the natural logarithm of the base: \( \frac{d}{dx}[a^x] = a^x \ln(a) \).

In exercises like ours, after rewriting the base \( 3 \) using the natural logarithm, we differentiate the exponential function as if it were \( e^u \), where \( u \) incorporates all variable-dependent parts. This requires the use of the chain rule to accurately compute the derivative.

Calculus

Calculus is the branch of mathematics that explores changes, represented through derivatives and integrals. At its core, calculus deals with understanding rates of change and the accumulation of quantities.

In the context of derivatives, calculus provides the tools to determine how a function changes over an interval. This is crucial for solving problems involving motion, growth, and areas under curves, among others.

• Derivatives: These describe the rate at which a function is changing at any point. Calculating a derivative means finding another function that gives the slope of the tangent line at any point on the graph.
• Integrals: In contrast, integrals explore the area under a curve, accumulated over an interval of the function.

When you compute derivatives ahead of exams, understanding the physical interpretation - like speeding up or slowing down in physics problems - enhances comprehension.

Exercises like these expand your problem-solving toolkit, refining your ability to apply these principles to both real-world and theoretical situations.