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Chapter 2: Problem 32

Find the indicated derivatives. If \(f(x)=x^{4},\) find \(f^{\prime}(-3)\).

Short Answer

Expert verified
The derivative at \( x = -3 \) is \( -108 \).

Step by step solution

01

Differentiate the Function

To find the derivative of the function, we use the power rule. The function is given as \( f(x) = x^4 \). The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \). Applying the power rule: \[ f'(x) = 4 \cdot x^{4-1} = 4x^3 \] This is the derivative of the function.
02

Evaluate the Derivative at the Given Point

Now, we need to find \( f'(-3) \) using the derivative we found in the previous step, \( f'(x) = 4x^3 \). Substitute \( x = -3 \) into the derivative: \[ f'(-3) = 4(-3)^3 \] Calculate \((-3)^3\): \[ (-3)^3 = -27 \] So, \[ f'(-3) = 4 \times (-27) = -108 \].Thus, the derivative of the function at \( x = -3 \) is \( -108 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is an essential concept in calculus, particularly in differentiation. It is a straightforward method to find the derivative of power functions. For a function of the form \( f(x) = x^n \), where \( n \) is any real number, the power rule helps us find its derivative easily. The rule states that the derivative \( f'(x) \) is given by: \[ f'(x) = n \cdot x^{n-1} \]To apply the power rule, simply follow these steps:
  • Identify the exponent \( n \) in the function \( x^n \).
  • Multiply the function by the exponent \( n \).
  • Reduce the exponent by 1, resulting in \( x^{n-1} \).
Using the power rule significantly simplifies the differentiation process, making it quick to find the derivative of functions like \( f(x) = x^4 \), resulting in \( f'(x) = 4x^3 \).
Derivative Evaluation
Derivative evaluation involves calculating the derivative of a function at a specific value of \( x \). In our example, we need to evaluate the derivative of \( f(x) = x^4 \), which is \( f'(x) = 4x^3 \), at \( x = -3 \).Evaluation steps include:
  • Substitute the value of \( x \) into the derivative \( f'(x) \).
  • Perform the necessary arithmetic calculations.
For our function, we substituted \( x = -3 \) into \( 4x^3 \): \[ f'(-3) = 4(-3)^3 \]Calculating yields \((-3)^3 = -27\), leading to:\[ f'(-3) = 4 \times (-27) = -108 \]Thus, the derivative of the function at \( x = -3 \) is \( -108 \). Evaluating derivatives at given points allows us to understand the rate of change of a function at specific intervals.
Mathematics Problem Solving
Mathematics problem solving is the application of mathematical methods to understand and solve problems. Differentiation problems usually require systematic steps, such as identifying the function form, applying differentiation rules, and evaluating results.Here's a simple guide to solving calculus problems using differentiation:
  • Understand the problem: Identify what you need to find鈥攊n this case, a derivative at a certain point.
  • Choose the right rule: Use the power rule for polynomials to find the derivative.
  • Calculate systematically: Differentiate first, then evaluate the derivative at the given value of \( x \).
By breaking down the problem into these steps, calculus problems become more manageable. Problem-solving in calculus often involves recognizing function patterns and applying the correct rules, which leads you to accurate results efficiently. This methodical approach enhances understanding and builds a strong foundation in mathematical problem solving.

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Most popular questions from this chapter

Use the Quotient Rule to find a general expression for the marginal average revenue. That is, calculate \(\frac{d}{d x}\left[\frac{R(x)}{x}\right]\) and simplify your answer.

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