Problem 79 - Calculus - AP Edition

Question

Find \(\frac{d}{d x}[f(x)]\) if \(\frac{d}{d x}[f(3 x)]=6 x\)

Step-by-Step Solution

Answer

The derivative \( f'(x) = \frac{2}{3}x \).

Step 1: Understand the Given Derivative

You're given that \( \frac{d}{d x}[f(3x)] = 6x \). This means when you take the derivative of \( f(3x) \) with respect to \( x \), you get \( 6x \).

Step 2: Apply the Chain Rule

To differentiate \( f(3x) \) with respect to \( x \), use the chain rule: \( \frac{d}{dx}[f(3x)] = f'(3x) \cdot 3 \). This results from treating \( f(3x) \) as \( f(u) \) where \( u = 3x \). Therefore, the derivative is \( f'(3x) \cdot 3 \).

Step 3: Set Up the Equation

You know that \( \frac{d}{d x}[f(3x)] = 6x \). From Step 2, substitute in the chain rule application: \( f'(3x) \cdot 3 = 6x \).

Step 4: Solve for \( f'(3x) \)

Solve the equation \( f'(3x) \cdot 3 = 6x \) by dividing both sides by 3: \( f'(3x) = 2x \).

Step 5: Find \( f'(x) \) using Substitution

To find \( f'(x) \), notice that \( f'(3x) = 2x \) means the derivative \( f' \) function will be affected by \( 3x \) substitution, therefore replace \( 3x \) by \( x \): effectively, this implies \( g'(x/3) = 2 \frac{x}{3} = \frac{2x}{3} \). For simplicity, just set \( u = x/3 \) back, making \( f'(u) = 2u \) i.e \( f'(x) = \frac{2}{3}x \) translating original \( x \) into \( u \).

Step 6: Show the Result for \( f'(x) \)

Thus, we find the derivative of \( f(x) \) with respect to \( x \) as \( f'(x) = \frac{2}{3}x \).

Key Concepts

Understanding the Chain Rule

The chain rule is an essential concept in calculus, particularly when differentiating composite functions—functions composed of other functions. It helps find the derivative of a function like \( f(g(x)) \), where \( f(x) \) and \( g(x) \) are two separate functions. The rule states that to differentiate \( f(g(x)) \) with respect to \( x \), you first differentiate \( f \) with respect to \( g(x) \), and then multiply by the derivative of \( g(x) \) with respect to \( x \).

The formula for the chain rule is:

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

In this exercise, the function \( f(3x) \) can be viewed as \( f(u) \) with the substitution of \( u = 3x \). So, when we apply the chain rule, we differentiate \( f(u) \) with respect to \( u \), then multiply by \( 3 \) (since \( u = 3x \), \( \frac{du}{dx} = 3 \)).

Basics of Derivatives

Derivatives represent how a function changes as its input changes, which is a fundamental idea in calculus. Essentially, the derivative gives us the rate at which a function is changing at any point. It is often thought of as the "slope" of the function at a specific point.

Consider a function \( y = f(x) \). The derivative of \( f \) with respect to \( x \) is noted as \( f'(x) \) or \( \frac{dy}{dx} \). This derivative tells us how \( y \) changes in response to changes in \( x \).

In simple terms, if you have a graph of the function, the derivative at any point gives you the slope of the tangent line at that point, which tells you how steep the graph is. The chain rule helps find derivatives for more complex functions, such as those involving other functions inside them.

The Function of a Function Concept

The "function of a function" idea is central to understanding composite functions like \( f(g(x)) \). Here, you have one function nested inside another. This is common in many real-world scenarios where the output of one process becomes the input to another.

To break it down, imagine a function \( g(x) \) that represents a process. A second function \( f(u) \), with \( u = g(x) \), takes the output of \( g \) as its input. This nested relationship calls for careful handling when performing calculus operations like differentiation.

For example, in this exercise, \( f(3x) \) is treated like \( f(u) \) such that \( u = 3x \). Understanding the function of a function is crucial in applying the chain rule effectively and requires recognizing how these nested functions interact with each other.