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Chapter 1: Problem 50

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

Short Answer

Expert verified
-\(\frac{3}{x^4}\)

Step by step solution

01

Identify the function

The function given is in the form of a power of x: \(f(x) = x^{-3}\).
02

Use the power rule of differentiation

The power rule states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
03

Apply the power rule

Using the power rule, take the derivative of \(f(x)\): \[\frac{d}{dx}(x^{-3}) = -3x^{-3-1} = -3x^{-4}\].
04

Simplify the result

Rewriting \(x^{-4}\) as a fraction: \[\frac{d}{dx}(x^{-3}) = -\frac{3}{x^4}\].

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Key Concepts

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

derivative
In mathematics, the derivative measures how a function changes as its input changes. Imagine you are hiking up a hill, and the hill's steepness at any point can be thought of as the derivative of the hill's height with respect to your horizontal position. We represent the derivative of a function using the notation \(\frac{d}{dx}\), which specifically shows how the function changes at every point.
power rule
The power rule is a fundamental shortcut in calculus to find the derivative of functions that are powers of x. If you have a function in the form \(f(x) = x^n\), the power rule says you can find its derivative, \(f'(x)\), by multiplying the function by the power and then subtracting one from the power. Mathematically, this is shown as: \[f(x) = x^n \rightarrow f'(x) = nx^{n-1}\] For example, if you have \(f(x) = x^3\), using the power rule, you get \[f'(x) = 3x^{3-1} = 3x^2\] This rule speeds up the differentiation process tremendously.
differentiation process
Differentiation is the process of finding the derivative of a function. The derivative tells you the rate at which a function is changing at any point. For our example, the differentiation process involves several clear steps:
  • Identify the function form
  • Apply the power rule
  • Simplify the result
In the given exercise, we identify the function as \(f(x) = x^{-3}\), then apply the power rule to find its derivative. We end up with the simpler form: \[f'(x) = -\frac{3}{x^4}\]
calculus basics
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Two of the main concepts in calculus are differentiation and integration.
Differentiation deals with finding the rate of change of a quantity, while integration is about finding the total size or value, like areas under curves. By mastering the basics of differentiation, such as the power rule, you build a strong foundation to tackle more advanced topics in calculus, making sense of real-world problems and changes.

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Most popular questions from this chapter

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { for } 0 \leq x<1 \\ 1 & \text { for } x=1 \\ 2 x-2 & \text { for } x>1\end{array}\right.\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{(1+h)^{2}-1}{h}\)

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

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Suppose that \(f(t)=3 t+2-\frac{12}{t}\) (a) What is the average rate of change of \(f(t)\) over the interval 2 to 3 ? (b) What is the (instantaneous) rate of change of \(f(t)\) when \(t=2 ?\)
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