Chapter 2: Q. 46 (page 222)

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {\ln (- x), if x < 0; \ln x, if x \geq 0}$

Short Answer

Expert verified

The derivative of function is

$f (x) = \frac{1}{x}, x < 0 D N E, x = 0 \frac{1}{x}, x > 0$

Step by step solution

Step 1. Given Information

The given function is $f (x) = {\ln (- x), if x < 0; \ln x, if x \geq 0}$

Step 2. Calculation

From the differentiation rules,

$\frac{d}{d x} (\ln (- x)) = \frac{1}{x} \frac{d}{d x} (\ln x) = \frac{1}{x} Let g (x) = \ln (- x), h (x) = \ln (x) And g' (x) = \frac{1}{x}, h' (x) = \frac{1}{x} When x = 0 then, g' (0) = h' (0) = undefined$

It follows that the function is not continuous at $x = 0$ .

Thus, the derivative of function is,

$f (x) = \frac{1}{x}, x < 0 D N E, x = 0 \frac{1}{x}, x > 0$

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {\ln (- x), if x < 0; \ln x, if x \geq 0}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.f(x)={ln(-x),ifx<0;lnx,ifx≥0}

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = {\ln (- x), if x < 0; \ln x, if x \geq 0}$