Problem 33 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. $$ \text { If } f \text { and } g \text { are differentiable at } x=2, \text { then } $$$$ \left.\frac{d}{d x}[f(x)-8 g(x)]\right|_{x=2}=f^{\prime}(2)-8 g^{\prime}(2) $$

Step-by-Step Solution

Answer

True, the statement correctly uses linearity and constants in differentiation.

Step 1: Understand the Problem Statement

We are given a statement about the differentiation of a function created from two differentiable functions \( f \) and \( g \). We need to determine if the derivative of the expression \( f(x) - 8g(x) \) evaluated at \( x=2 \) is equal to \( f'(2) - 8g'(2) \).

Step 2: Apply Differentiation Rules

Recall the differentiation rule for a constant multiplied by a function: \( \frac{d}{dx}[c \cdot h(x)] = c \cdot h'(x) \). Also, remember the linearity property of derivatives: \( \frac{d}{dx}[a + b] = a'(x) + b'(x) \).

Step 3: Differentiate the Function

Apply the differentiation rule to the function \( f(x) - 8g(x) \). This gives \( \frac{d}{dx}[f(x) - 8g(x)] = f'(x) - 8g'(x) \) because you differentiate each term separately and apply the constant multiplier to the derivative of \( g(x) \).

Step 4: Evaluate the Derivative at x=2

Substitute \( x = 2 \) into the differentiated expression: \( f'(2) - 8g'(2) \).

Step 5: Compare Both Sides of the Statement

The expression derived from applying differentiation and the expression provided in the statement, \( f'(2) - 8g'(2) \), are equivalent. Thus, the given statement matches the result from applying differentiation rules.

Key Concepts

Linearity of Derivatives

The principle of the linearity of derivatives simplifies the differentiation process, especially when dealing with functions that are a sum or difference of other functions. When we say derivatives are linear, it means you can differentiate each part of the function independently. For instance, if you have a function formed by adding or subtracting two other functions, the derivative of the whole function is simply the sum or difference of the individual derivatives.

In the original exercise, the function given is:

• \( f(x) - 8g(x) \)

The linearity rule tells us that, when differentiating, each component of the expression can be treated separately. This leads us directly to:

• \( \frac{d}{dx}[f(x) - 8g(x)] = f'(x) - 8g'(x) \)

This property is incredibly useful, as it allows us to break down complex problems into manageable parts, dealing with one function at a time.

Derivative of a Constant Multiple

The differentiation rule for a constant multiple is a cornerstone of calculus. It states that when you have a constant multiplied by a function, you can take the constant out of the derivative. Mathematically, for a constant \( c \) and a function \( h(x) \), the rule is:

• \( \frac{d}{dx}[c \, h(x)] = c \, h'(x) \)

In simple terms, this means if you're differentiating a function that's being scaled by a constant factor, you find the derivative of the function first, then multiply it by the constant.

Applying this to our exercise, for the term \(-8g(x)\), you apply the rule:

• \( \frac{d}{dx}[-8g(x)] = -8 \cdot g'(x) \)

Here, \(-8\) is the constant that scales the function \( g(x) \). When you differentiate, you maintain this scaling factor but focus on the change in \( g(x) \). This constant multiplier rule makes handling such terms straightforward and mechanical, reducing the chance of errors.

Evaluating Derivatives at a Point

After differentiating a function, the next step often involves evaluating its derivative at a specific point. This tells us the rate of change at that particular point. Given a function's derivative \( f'(x) \), evaluating it at \( x = a \) simply involves substituting \( x \) with \( a \).

In the original step-by-step solution, the task was to evaluate the derivatives at \( x = 2 \). Once you derive:

• \( f'(x) - 8g'(x) \)

you substitute \( x = 2 \) to get:

• \( f'(2) - 8g'(2) \)

This substitution doesn't change the process; it just provides a numerical result that corresponds to the slope or rate of change at the point where \( x = 2 \). This tells you how sharply the function is rising or falling, which is often crucial for understanding the behavior of mathematical models or systems at particular points.