Problem 5

Question

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

Step-by-Step Solution

Verified

Answer

The derivative is $D_x y = -4x^{-3}$.

1Step 1: 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.

2Step 2: 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 $n = -2$.

3Step 3: Compute the derivative

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

4Step 4: Simplify the expression

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

Key Concepts

Power RuleDifferentiationCalculus

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.

Problem 5

Problem 6

Other exercises in this chapter

Problem 5

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sec x=1 / \cos x $$

View solution

Problem 5

Use $f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h$ to find the derivative at $x$. $$ s(x)=2 x+1 $$

View solution

Problem 6

Find $d y$. $$ y=(\tan x+1)^{3} $$

View solution

Problem 6

Find $D_{x} y$. $$ y=\sinh \left(x^{2}+x\right) $$

View solution