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 and Rule to Apply

The given function is $y = 2x^{-2}$. It's a power function, and we need to use the power rule for differentiation.

2Step 2: Recall the Power Rule

The power rule states: If $y = ax^n$, then the derivative $D_{x} y = anx^{n-1}$. In this, $a$ is a constant, and $n$ is the power to which $x$ is raised.

3Step 3: Identify $a$ and $n$

For the function $y = 2x^{-2}$, the constant $a = 2$ and the exponent $n = -2$.

4Step 4: Apply the Power Rule

Using the power rule, multiply the exponent $n$ by the constant $a$ and reduce the exponent by one: $D_{x} y = 2(-2)x^{-2-1}$.

5Step 5: Simplify the Expression

Simplify the multiplication and reduce the exponent: $D_{x} y = -4x^{-3}$. This is the derivative of the given function.

Key Concepts

Power RuleDerivativePower Function

Power Rule

Differentiation is a key concept in calculus, and understanding the "power rule" can make the process much easier. The power rule is a fundamental tool used when dealing with derivatives of functions, specifically power functions. It applies to expressions in the form of

$y = ax^n$

where $a$ is a constant and $n$ is any real number exponent. The power rule formula is simple:

$D_{x} y = anx^{n-1}$

This essentially means you take the exponent $n$, multiply it by the constant $a$, and then decrease the exponent by one.
This rule saves time and minimizes errors compared to the lengthy process of applying the definition of a derivative directly.

Derivative

The "derivative" of a function is a measure of how the function changes as its input changes. In simpler terms, it gives us the slope of the function's graph at any given point. Understanding derivatives is crucial, as they form the backbone of calculus applications in many scientific fields.For a function $y = f(x)$, the derivative is represented by $D_{x} y$ or $\frac{dy}{dx}$. It tells us how the function $y$ reacts to small changes in $x$.

In our exercise, we've found that the derivative of the function $y = 2x^{-2}$ is $D_{x} y = -4x^{-3}$.

This new function represents the rate of change of $y$ with respect to changes in $x$. It also illustrates how the original curve behaves, indicating whether it slopes upwards or downwards as $x$ increases.

Power Function

A "power function" is a type of mathematical expression where a variable is raised to a constant power. These functions are quite simple in structure but can represent a variety of complex behaviors, especially when dealing with calculus problems. A general power function is represented as:

$y = ax^n$

where $a$ is a constant coefficient, $x$ is the variable, and $n$ denotes the power. These functions can take many forms, including polynomial expressions or even basic algebraic functions.In the context of our exercise, $y = 2x^{-2}$ is a power function where $a = 2$ and $n = -2$. The negative exponent implies that the function represents a reciprocal power, introducing interesting behaviors such as asymptotes. These are lines that the graph approaches but never touches. Understanding the concept of a power function allows you to better tackle derivative problems using rules like the power rule effectively.

Problem 5

Other exercises in this chapter

Problem 5

Find $d y$. $$ y=(\sin x+\cos x)^{3} $$

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Problem 5

Assuming that each equation defines a differentiable function of $x$, find $D_{x}$ y by implicit differentiation. $$ x y^{2}=x-8 $$

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Problem 5

Find $D_{x} y .$ $y=\left(x^{3}-2 x^{2}+3 x+1\right)^{11}$

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Problem 5

In Problems $5-29$, find the indicated derivative by using the rules that we have developed. $$ D_{x}\left(3 x^{5}\right) $$

View solution