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Chapter 1: Problem 45

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

Short Answer

Expert verified
\( f^{\text{ ' }}(x) = \frac{1}{2\sqrt{x+2}} \).

Step by step solution

01

Recall the definition of the derivative

The derivative of a function is defined as \[ f^{\text { ' }}(x) = \frac{d}{dx} f(x) = \frac{f(x+h) - f(x)}{h} \bigg|_{h \to 0}. \]
02

Substitute the given function into the definition

Given \( f(x) = \sqrt{x+2} \), we need to substitute this into the limit definition:\[ f^{\text { ' }}(x) = \frac{ \sqrt{(x+h)+2} - \sqrt{x+2} }{h} \bigg|_{h \to 0}. \]
03

Simplify the numerator using conjugates

Multiply the numerator and the denominator by the conjugate of the numerator:\[ f^{\text{ ' }}(x) = \frac{\left(\sqrt{x+h+2} - \sqrt{x+2} \right) \cdot \left(\sqrt{x+h+2} + \sqrt{x+2} \right)}{h \cdot \left(\sqrt{x+h+2} + \sqrt{x+2} \right)}.\]This simplifies to:\[ f^{\text{ ' }}(x) = \frac{(x+h+2) - (x+2)}{h \cdot \left(\sqrt{x+h+2} + \sqrt{x+2} \right)}.\]
04

Simplify the fraction

Continuing from the expression:\[ f^{\text{ ' }}(x) = \frac{(x+h+2) - (x+2)}{h\left(\sqrt{x+h+2}+\sqrt{x+2}\right)}\]The numerator simplifies to:\[ f^{\text{ ' }}(x) = \frac{h}{h \left(\sqrt{x+h+2}+\sqrt{x+2}\right)}.\]Cancel out the \(h\):\[ f^{\text{ ' }}(x) = \frac{1}{\sqrt{x+h+2} + \sqrt{x+2}}. \]
05

Take the limit as h approaches zero

Take the limit of the simplified formula as \(h\) approaches zero:\[ f^{\text{ ' }}(x) = \frac{1}{\sqrt{x+2} + \sqrt{x+2}} = \frac{1}{2\sqrt{x+2}}. \]

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Key Concepts

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

Limit Definition
The limit definition of a derivative is a fundamental concept in calculus. It provides a precise way to find the slope or rate of change of a function at any given point. This definition is written as \(f^{\prime}(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\). Here, \(h\) represents a small increment in \(x\), and we observe how the function changes over this tiny interval.
Starting with this definition allows us to calculate derivatives for different functions. By plugging in the given function into this formula, we can analyze and break down the function's behavior as \(h\) approaches zero.
Function Derivative
Finding the derivative of a function means determining its instantaneous rate of change. For the function given, \(f(x) = \sqrt{x+2}\), we follow several steps:
  • We substitute \(f(x)\) into the limit definition formula: \(\frac{\sqrt{(x+h)+2} - \sqrt{x+2}}{h}\).
  • Next, we multiply by the conjugate to simplify the expression: \(\frac{(\sqrt{x+h+2} - \sqrt{x+2})(\sqrt{x+h+2} + \sqrt{x+2})}{h(\sqrt{x+h+2} + \sqrt{x+2})}\).
  • This helps reduce the complexity of the radicals, eventually leading to the simplified form \(\frac{h}{h(\sqrt{x+h+2} + \sqrt{x+2})}\).
  • We then cancel out \(h\) and proceed to take the limit as \(h\) approaches zero, giving us the final derivative: \(f^{\prime}(x) = \frac{1}{2\sqrt{x+2}}\).
These steps illustrate the critical path from a basic function to its derivative.
Simplifying Radicals
Simplifying radicals is a crucial step when dealing with complex algebraic expressions, especially in the context of derivatives.
  • We use the concept of conjugates, which involves multiplying the numerator and the denominator by the conjugate of the numerator. The conjugate of \(\sqrt{a} - \sqrt{b}\) is \(\sqrt{a} + \sqrt{b}\).
  • This approach helps eliminate the square roots, simplifying the expression to a more manageable form, like turning \( a - b \) back into basic terms.
  • For example, in our problem: \(\frac{(\sqrt{x+h+2} - \sqrt{x+2})(\sqrt{x+h+2} + \sqrt{x+2})}{h(\sqrt{x+h+2} + \sqrt{x+2})}\) simplifies to \(\frac{(x+h+2) - (x+2)}{h(\sqrt{x+h+2} + \sqrt{x+2})}\), which further reduces to \(\frac{h}{h(\sqrt{x+h+2} + \sqrt{x+2})}\).
This simplification is key in progressing to the step where the \(h\) terms cancel out and leads us to the final result.

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Most popular questions from this chapter

A particle is moving in a straight line in such a way that its position at time \(t\) (in seconds) is \(s(t)=t^{2}+3 t+2\) feet to the right of a reference point, for \(t \geq 0.\) (a) What is the velocity of the object when the time is 6 seconds? (b) Is the object moving toward the reference point when \(t=6 ?\) Explain your answer. (c) What is the object's velocity when the object is 6 feet from the reference point?

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.

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

Let \(y\) denote the average amount claimed for itemized deductions on a tax return reporting \(x\) dollars of income. According to Internal Revenue Service data, \(y\) is a linear function of \(x .\) Moreover, in a recent year income tax returns reporting \(\$ 20,000\) of income averaged \(\$ 729\) in itemized deductions, while returns reporting \(\$ 50,000\) averaged \(\$ 1380 .\) (a) Determine \(y\) as a function of \(x.\) (b) Graph this function in the window \([0,75000]\) by \([0,2000].\) (c) Give an interpretation of the slope in applied terms. (d) Determine graphically the average amount of itemized deductions on a return reporting \(\$ 75,000.\) (e) Determine graphically the income level at which the average itemized deductions are \(\$ 1600 .\) (f) If the income level increases by \(\$ 15,000,\) by how much do the average itemized deductions increase?
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