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Chapter 2: Q 11. (page 238)

Find the derivatives of the function: $f (x) = \frac{1 - \frac{1}{x}}{\sqrt{x}} .$

Short Answer

Expert verified

The derivative of $f (x) = \frac{1 - \frac{1}{x}}{\sqrt{x}} i s \frac{- x + 3}{2 x^{2} \sqrt{x}} .$

Step by step solution

01

Step 1 / 2Step 1: Given information

The given expression is $f (x) = \frac{1 - \frac{1}{x}}{\sqrt{x}} .$

02

Simplification

$f' (x) = \frac{d}{d x} (\frac{1 - \frac{1}{x}}{\sqrt{x}}) = \frac{d}{d x} (\frac{\frac{x - 1}{x}}{\sqrt{x}}) = \frac{d}{d x} (\frac{x - 1}{x \sqrt{x}}) = \frac{d}{d x} (\frac{x}{x \sqrt{x}} - \frac{1}{x \sqrt{x}}) = \frac{d}{d x} (x^{- \frac{1}{2}}) - \frac{d}{d x} (x^{- \frac{3}{2}}) = - \frac{1}{2 x^{\frac{3}{2}}} + \frac{3}{2 x^{- \frac{5}{2}}} = \frac{- x + 3}{2 x^{2} \sqrt{x}} H e n c e, f' (x) = \frac{- x + 3}{2 x^{2} \sqrt{x}} .$

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Most popular questions from this chapter

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

The line tangent to the graph of $y = 4 x + 3$ at the point $(- 2, - 5)$

In Exercises 69–80, determine whether or not $f$ is continuous and/or differentiable at the given value of $x$ . If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

$f (x) = \{\begin{cases} x^{2}, i f x \leq 1 \\ 2 x - 1, i f x > 1, \end{cases} x = 1$

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = 3 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{2}{3}}$

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = {(x \sqrt{x + 1})}^{- 2}$

Velocity $v (t)$ is the derivative of position $s (t)$ . It is also true that acceleration $a (t)$ (the rate of change of velocity) is the derivative of velocity. If a race car’s position in miles t hours after the start of a race is given by the function $s (t)$ , what are the units of $s (1.2)$ ? What are the units and real-world interpretation of $v (1.2)$ ? What are the units and real-world interpretations of $a (1.2)$ ?

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