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Chapter 1: Problem 18

Find \(f^{\prime \prime}(x)\). $$f(x)=2 x^{-2}$$

Short Answer

Expert verified
12 x^{-4}

Step by step solution

01

- Rewrite the Function

Rewrite the given function for easier differentiation. The function is given as: \[ f(x) = 2 x^{-2} \]
02

- Differentiate the Function to Find the First Derivative

Apply the power rule for differentiation, which states that \( f'(x) = nx^{n-1} \). For the given function \( f(x) = 2 x^{-2} \), we get: \[ f'(x) = 2 \times (-2) x^{-2-1} = -4 x^{-3} \]
03

- Differentiate Again to Find the Second Derivative

Apply the power rule again to the first derivative \( f'(x) = -4 x^{-3} \). We get: \[ f''(x) = -4 \times (-3) x^{-3-1} = 12 x^{-4} \]

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which is a measure of how the function's value changes as its input changes. The process of differentiation calculates the slope of the function at any given point.

There are many rules and techniques to differentiate functions effectively. In this exercise, we focus on the power rule. Differentiation helps in various fields like physics, engineering, and economics to analyze and understand rate changes.

For instance, if we know the position of an object over time, differentiation allows us to find its velocity and acceleration, providing deeper insights into its motion.
Power Rule
The power rule is one of the simplest and most commonly used rules in differentiation. It applies to functions of the form \[ f(x) = x^n \].

According to the power rule:
\[ f'(x) = nx^{n-1} \]

Suppose we have a function \[ f(x) = 2x^{-2} \]. To find its derivative, we apply the power rule:

  • First step: Multiply the coefficient (2) by the exponent (-2).
  • Second step: Decrease the exponent (-2) by 1.


Thus, we get: \[ f'(x) = 2 \times (-2) x^{-2-1} = -4 x^{-3} \]

We use the power rule again to find the second derivative from the first derivative:

\[ f''(x) = -4 \times (-3)x^{-3-1} = 12x^{-4}. \]
Calculus
Calculus is a branch of mathematics that studies how things change. It consists of two main parts: differentiation and integration. While differentiation deals with the rate of change, integration focuses on the accumulation of quantities.

Calculus is foundational for understanding much of modern science and engineering. It allows us to model and solve problems involving changing systems, from simple physical motions to complex biological processes.

In our exercise, the differentiation of \[ f(x) = 2x^{-2} \] using the power rule provides insights into how the function behaves as x changes. It's a fundamental skill for students learning to analyze real-world systems.

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Most popular questions from this chapter

Differentiate. $$g(x)=\sqrt{\frac{x^{2}-4 x}{2 x+1}}$$

Differentiate each function. $$f(t)=\left(t^{5}+3\right) \cdot \frac{t^{3}-1}{t^{3}+1}$$

Differentiate. $$y=\sqrt{(2 x-3)^{2}+1}$$

Use the derivative to help show whether each function is always increasing, always decreasing, or neither. $$f(x)=x^{5}+x^{3}$$

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative.. If \(f(x)=\sqrt[3]{x},\) find \(\frac{d}{d x}[(f \circ f \circ f)(x)]\).
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