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Chapter 2: Q. 48 (page 222)

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

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

01

Step 1. Given Information

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

02

Step 2. Calculation

From the differentiation rules,

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

03

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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Most popular questions from this chapter

Use (a) the $h 鈫� 0$ definition of the derivative and then

(b) the $z 鈫� c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23鈥�38.

26. $f (x) = \frac{1}{x^{2}}, x = 2$

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

$f' (x) = {(3 x + 1)}^{3}, f (2) = 1$

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

$f' (x) = 7 x^{2} + 8 x^{11} - 18; f (0) = - 2$

Suppose f is ant cubic polynomial function $f (x) = a x^{3} + b x^{2} + c x + d$ prove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2

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

$f (x) = \sqrt{2 - \sqrt{3 x + 1}}$

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