Problem 33 - Calculus - AP Edition

Question

Determine whether the statement is true or false. Explain your answer. If \(f\) and \(g\) are differentiable at \(x=2,\) 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

The statement is true; the differentiation rules validate the equality.

Step 1: Understand the Problem

We need to verify if the derivative of the function \( h(x) = f(x) - 8g(x) \) with respect to \( x \) at \( x = 2 \) is equal to \( f'(2) - 8g'(2) \). We assume \( f \) and \( g \) are differentiable at \( x = 2 \).

Step 2: Apply the Derivative Rules

Using the linearity of the derivative, the derivative of \( h(x) = f(x) - 8g(x) \) is \( h'(x) = f'(x) - 8g'(x) \). This is derived by using the constant multiple rule and the sum/difference rule for derivatives.

Step 3: Evaluate the Derivative at x = 2

Substituting \( x = 2 \) into the derivative expression, we find \( h'(2) = f'(2) - 8g'(2) \).

Step 4: Compare with Given Expression

The problem states that we need \( \frac{d}{dx}[f(x) - 8g(x)] \big|_{x=2} = f'(2) - 8g'(2) \). The derived expression matches this, meaning the statement is correct.

Key Concepts

Derivative Rules

When we talk about derivative rules, we're exploring the basic instructions or guidelines that help us find the derivative of a function. At the heart of calculus, derivatives measure how a function changes as its input changes. Derivative rules simplify this process, allowing us to find these rates of change without starting from scratch each time.

• The power rule: This helps when dealing with polynomials.
• The product rule: Useful when multiplying two functions.
• The quotient rule: Applicable when dividing one function by another.

These rules streamline the process, saving time and reducing potential mistakes. For example, in this exercise, knowing how to apply the sum/difference rule and the constant multiple rule allows us to determine the derivative with ease.

Linearity of the Derivative

The linearity of the derivative is a fundamental principle that states the derivative of a sum or difference of functions is the sum or difference of their derivatives. This property makes complex problems more manageable by breaking down expressions into simpler parts. In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), their derivative can be expressed as:\[(f(x) + g(x))' = f'(x) + g'(x)\]Similarly:\[(f(x) - g(x))' = f'(x) - g'(x)\]This linearity is key in transforming the given expression \( rac{d}{dx}[f(x) - 8g(x)] \) to the simpler form \( f'(x) - 8g'(x) \) using the sum/difference rule.

Constant Multiple Rule

The constant multiple rule is a straightforward yet powerful tool in calculus. It states that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function. This rule simplifies calculations considerably because it allows us to "factor out" constants. Formally, if \( c \) is a constant and \( f(x) \) is a function, then:\[(c \, f(x))' = c \, f'(x)\]In the given exercise, this rule is crucial for simplifying \(-8g(x)\). We apply the constant multiple rule to get \(-8g'(x)\), maintaining the constant alongside the derivative. This approach makes working with derivatives more efficient and less error-prone.

Sum/Difference Rule

The sum/difference rule is another fundamental derivative rule that helps us deal with addition or subtraction of functions effectively. This rule tells us that the derivative of two added or subtracted functions is the same as adding or subtracting their individual derivatives, a principle related closely to linearity. To express it clearly:

• If you have \( (f(x) + g(x))' = f'(x) + g'(x) \)
• Or, \( (f(x) - g(x))' = f'(x) - g'(x) \)

In the context of the problem, applying the sum/difference rule helps rewrite the derivative of \( f(x) - 8g(x) \) into a readable formula: \( f'(x) - 8g'(x) \). This rule simplifies processes, allowing derivatives to be found swiftly and accurately.