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Chapter 1: Problem 54

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=x^{-4}\)

Short Answer

Expert verified
The derivative is \( \frac{d y}{d x} = -4 x^{-5} \).

Step by step solution

01

- Understand the Power Rule

The power rule states that if you have a function of the form \[ y = x^n \] then the derivative with respect to x is \[ \frac{d y}{d x} = n x^{n-1} \].
02

- Identify the Components

In the given function, \[ y = x^{-4} \] we have: \[ n = -4 \].
03

- Apply the Power Rule

Using the power rule, substitute \[ n = -4 \] into the formula: \[ \frac{d y}{d x} = -4 x^{-4-1} = -4 x^{-5} \].

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Key Concepts

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

Understanding Derivatives
A derivative represents the rate at which a function is changing at any given point. Think of it as a way to measure how a small change in one variable affects another variable. In mathematical terms, the derivative of a function \( y = f(x) \) is denoted as \( \frac{d y}{d x} \). For example, in the exercise mentioned, we're asked to find the derivative of \( y = x^{-4} \). This process involves understanding both the concept of derivatives and applying relevant rules.
The Process of Differentiation
Differentiation is the process of finding a derivative. It's like following a recipe to get the final dish. Here are some basic steps:
  • Identify the function you need to differentiate.
  • Apply differentiation rules such as the power rule, product rule, quotient rule, etc.
  • Simplify the resulting expression, if possible.
In our case, the function is \( y = x^{-4} \). We use the power rule, which simplifies the differentiation process significantly.
Applying the Power Rule
The power rule is one of the most fundamental rules in calculus. It states that if \( y = x^n \), then \( \frac{d y}{d x} = n x^{n-1} \). To apply this rule:
  • Identify the exponent \( n \) in the given function \( y = x^n \).
  • Subtract 1 from the exponent.
  • Multiply the entire expression by the original exponent.
For the exercise \( y = x^{-4} \):
  • Exponent \( n = -4 \).
  • Subtract 1 from \( n \) to get \( -5 \).
  • Multiply by the original exponent: \( -4 x^{-5} \).
So, the derivative of \( y = x^{-4} \) is \( \frac{d y}{d x} = -4 x^{-5} \).
Putting It All Together
Using these concepts together helps in finding derivatives more easily. By understanding each step clearly:
  • Recognize the function form.
  • Apply the correct rule.
  • Simplify and solve.
Differentiation, derivatives, and specific rules like the power rule give us powerful tools for calculus problems, making complex processes simpler and more understandable.

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Most popular questions from this chapter

Compute the difference quotient $$ \frac{f(x+h)-f(x)}{h} . $$ Simplify your answer as much as possible. \(f(x)=x^{3}\) [Hint: \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\).]

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{x^{2}+x-12}{x+4}, x \neq-4\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=x+\frac{1}{x}\)

Let \(f(x)\) be the number of toys sold when \(x\) dollars are spent on advertising. Interpret the statements \(f(100,000)=3,000,000\) and \(f^{\prime}(100,000)=30\).

Find the indicated derivative. \(\frac{d}{d x}\left(x^{-3}\right)\)
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