Q. 70 Find the derivatives of each of ... [FREE SOLUTION]

Chapter 2: Q. 70 (page 198)

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = \{\begin{cases} 1, i f x 鈮� - 1 \\ x^{2 / 3}, i f x > - 1 \end{cases}$

Short Answer

Expert verified

The derivative of the function is $f' (x) = \{\begin{cases} 0, i f x < - 1 \\ D N E, i f x = - 1 \\ \begin{array}{rcl} \frac{2}{3} x^{\frac{- 1}{3}}, i f x & > & - 1 \end{array} \end{cases}$ .

Step by step solution

Step 1. Given Information

The given function is $f (x) = \{\begin{cases} 1, i f x 鈮� - 1 \\ x^{2 / 3}, i f x > - 1 \end{cases}$

Step 2. Find the derivative

It is known that, the power rule of derivative is $(x^{n})' = n x^{n - 1}$ and the derivative of the constant is 0.
Find the derivative for $x < - 1$ .

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (1) \\ = & 0 \end{array}$

Find the derivative for $x > - 1$ .

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (x^{2 / 3}) \\ = & \frac{2}{3} x^{\frac{2}{3} - 1} \\ = & \frac{2}{3} x^{\frac{- 1}{3}} \end{array}$

Since $0 鈮� \begin{array}{rcl} \frac{2}{3} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} 1 \end{array} \begin{array}{rcl} ) \end{array} \begin{array}{rcl} \end{array}$ ^-13, the derivatives at left and right pieces are not equal at $x = - 1$ . The derivative does not exists at $x = - 1$ .
So, the derivative of the function is, $f' (x) = \{\begin{cases} 0, i f x < - 1 \\ D N E, i f x = - 1 \\ \begin{array}{rcl} \frac{2}{3} x^{\frac{- 1}{3}}, i f x & > & - 1 \end{array} \end{cases}$ .

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91影视

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = \{\begin{cases} 1, i f x 鈮� - 1 \\ x^{2 / 3}, i f x > - 1 \end{cases}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the derivative

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91影视

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72. f(x)=1,ifx鈮�-1x2/3,ifx>-1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = \{\begin{cases} 1, i f x 鈮� - 1 \\ x^{2 / 3}, i f x > - 1 \end{cases}$