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Chapter 3: Problem 5

Find \(D_{x} y\) using the rules of this section. $$ y=2 x^{-2} $$

Short Answer

Expert verified
The derivative is \(D_x y = -4x^{-3}\).

Step by step solution

01

Identify the function format

The given function is \(y = 2x^{-2}\), which is in the format \(ax^n\) where \(a = 2\) and \(n = -2\). This is a power function.
02

Apply the power rule of differentiation

The power rule of differentiation states that if \(y = ax^n\), then the derivative \(D_x y = a \cdot n \cdot x^{n-1}\). In our case, \(a = 2\) and \(n = -2\).
03

Compute the derivative

Using the power rule, substitute \(a = 2\) and \(n = -2\) into the formula. This gives \(D_x y = 2 imes (-2) imes x^{-3}\).
04

Simplify the expression

Compute the product \(2 imes (-2)\) which equals \(-4\). So, the derivative is \(D_x y = -4x^{-3}\).

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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 a fundamental tool in calculus used for finding derivatives of power functions. It's a simple and powerful method that helps us understand how functions change. The rule states: if you have a function of the form \( y = ax^n \), where \( a \) is a constant, and \( n \) is a real number, then the derivative with respect to \( x \) is given by:
  • Multiply the exponent \( n \) by the constant \( a \).
  • Reduce the exponent by 1.
  • Express it as: \( D_x y = a \cdot n \cdot x^{n-1} \).
This rule makes differentiated functions easier to handle, especially when dealing with polynomials or more complex expressions.
In the exercise, the Power Rule was applied to \( y = 2x^{-2} \), identifying \( a = 2 \) and \( n = -2 \). By following the steps of the Power Rule, we derived \( D_x y = -4x^{-3} \).
Differentiation
Differentiation is a process in calculus that deals with finding the derivative of a function. The derivative describes the rate at which the function's value changes at any given point.Differentiation is essential in understanding motion, growth, and other changes. It's like finding the slope of a curve at any point, which indicates whether the function is increasing or decreasing.
  • Derivatives can tell you the steepness of the curve, which is crucial in many fields such as physics and economics.
  • Through differentiation, problems involving instantaneous rates of change can be solved.
In the given problem, differentiation was used to transform the function \( y = 2x^{-2} \) into its derivative \( D_x y = -4x^{-3} \), providing insight into how the original function behaves.
Calculus
Calculus is the branch of mathematics that studies change, emphasizing the concepts of differentiation and integration. It's a powerful tool used in various scientific fields to analyze dynamic systems and patterns. Within calculus:
  • Differentiation helps to find instantaneous rates of change.
  • Integration is the reverse process, dealing with accumulation.
Calculus allows us to describe the physical world, such as how fast an object is moving, or how much area is under a curve.
The exercise demonstrated a fundamental aspect of calculus: using differentiation to solve problems. By applying the power rule, students learn how calculus makes it possible to understand complex relationships and behavior of functions in a clear and structured way.

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