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Chapter 1: Problem 40

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\frac{1}{x^{2}}$$

Short Answer

Expert verified
The derivative of \( f(x) = \frac{1}{x^{2}} \) is \( f^{\textprime}(x) = \frac{-2}{x^{3}} \).

Step by step solution

01

Recall the Definition of a Derivative

The derivative of a function, denoted as \( f^{\textprime}(x) \), is defined by the limit: \[ f^{\textprime}(x) = \lim_{{h \rightarrow 0}} \frac{f(x+h) - f(x)}{h} \]. We will apply this definition to compute the derivative of \( f(x) = \frac{1}{x^{2}} \).
02

Plug Into the Definition

Using \( f(x) = \frac{1}{x^{2}} \), we find \( f(x+h) \): \[ f(x+h) = \frac{1}{(x+h)^{2}} \]. Now plug both \( f(x) \) and \( f(x+h) \) into the definition of the derivative: \[ f^{\textprime}(x) = \lim_{{h \rightarrow 0}} \frac{\frac{1}{(x+h)^{2}} - \frac{1}{x^{2}}}{h} \].
03

Simplify the Expression

Combine the fractions in the numerator: \[ f^{\textprime}(x) = \lim_{{h \rightarrow 0}} \frac{\frac{1}{(x+h)^{2}} - \frac{1}{x^{2}}}{h} = \lim_{{h \rightarrow 0}} \frac{x^{2} - (x+h)^{2}}{h \cdot x^{2} \cdot (x+h)^{2}} \]. Simplify the numerator: \[ x^{2} - (x+h)^{2} = x^{2} - (x^{2} + 2xh + h^{2}) = -2xh - h^{2} \]. So the expression becomes: \[ f^{\textprime}(x) = \lim_{{h \rightarrow 0}} \frac{-2xh - h^{2}}{h \cdot x^{2} \cdot (x+h)^{2}} \].
04

Cancel and Take the Limit

Factor \( h \) out of the numerator: \[ f^{\textprime}(x) = \lim_{{h \rightarrow 0}} \frac{h \cdot (-2x - h)}{h \cdot x^{2} \cdot (x+h)^{2}} = \lim_{{h \rightarrow 0}} \frac{-2x - h}{x^{2} \cdot (x+h)^{2}} \]. As \( h \rightarrow 0 \), the term \( -h \) goes to zero, so we get: \[ f^{\textprime}(x) = \frac{-2x}{x^{2} \cdot x^{2}} = \frac{-2}{x^{3}} \].

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Key Concepts

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

Derivative Definition
The derivative of a function gives us the rate at which the function's value changes with respect to its input.

Mathematically, the derivative of a function f(x) is defined by:
\[ f^{\prime}(x) = \lim_{{h \rightarrow 0}} \frac{f(x+h) - f(x)}{h} \]

This formula tells us how to find the gradient (or slope) of the function at any point x. It is essential to understand that finding the derivative using this definition involves computing a limit.

When applied to the given function, \[f(x) = \frac{1}{x^{2}}\], we use the definition of the derivative to find the derivative through a series of steps: substituting the function, plugging it into the formula, simplifying the expression, and finally taking the limit.

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Most popular questions from this chapter

Apply the three-step method to compute the derivative of the given function. $$f(x)=-x^{2}$$

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(1), \text { where } f(x)=\frac{1}{1+x^{2}}$$

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