Chapter 2: Q. 48 (page 222)

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = \{\frac{1}{2} - \frac{1}{4} x, x \leq 0; \frac{1}{1 + e^{x}}, x > 0\}$

Short Answer

Expert verified

The derivative of function is

$f' (x) = - \frac{1}{4}, x < 0 - \frac{1}{4}, x = 0 \frac{- e^{x}}{{(1 + e^{x})}^{2}}, x > 0$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \{\frac{1}{2} - \frac{1}{4} x, x \leq 0; \frac{1}{1 + e^{x}}, x > 0\}$

Step 2. Calculation

From the differentiation rules,

$\frac{d}{d x} (\frac{1}{2} - \frac{1}{4} x) = - \frac{1}{4} \frac{d}{d x} (\frac{1}{1 + e^{x}}) = \frac{- e^{x}}{{(+ e^{x})}^{2}} let g (x) = \frac{1}{2} - \frac{1}{4} x, h (x) = \frac{1}{1 + e^{x}} And g' (x) = - \frac{1}{4}, h' (x) = \frac{- e^{x}}{{(+ e^{x})}^{2}}$

Step 3. Calculation

$C h e c k g (0) = h (0) g (0) = \frac{1}{2} h (0) = \frac{1}{2}$

This is true which follows that the function is continuous at $x = 0$

$C h e c k g' (0) = h' (0) g' (0) = - \frac{1}{4} h' (0) = \frac{- e^{0}}{{(1 + e^{0})}^{2}} = - \frac{1}{4}$

So, although function is continuous at zero, it is always differentiable.

Thus, the derivative of the function is

$f' (x) = - \frac{1}{4}, x < 0 - \frac{1}{4}, x = 0 \frac{- e^{x}}{{(1 + e^{x})}^{2}}, x > 0$

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = \{\frac{1}{2} - \frac{1}{4} x, x \leq 0; \frac{1}{1 + e^{x}}, x > 0\}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48. f(x)=12-14x,x≤0;11+ex,x>0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.
$f (x) = \{\frac{1}{2} - \frac{1}{4} x, x \leq 0; \frac{1}{1 + e^{x}}, x > 0\}$