Chapter 2: Q. 46 (page 210)

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

Short Answer

Expert verified

The required answer is $- \frac{1}{4 \sqrt{1 + \sqrt{1 + \sqrt{1 - 2 x}}} \sqrt{1 + \sqrt{1 - 2 x}} \sqrt{1 - 2 x}}$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \sqrt{1 + \sqrt{1 + \sqrt{1 - 2 x}}}$

Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$f' (x) = \frac{d}{d x} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{1}{2}} = \frac{1}{2} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{1}{2} - 1} \frac{d}{d x} (1 + \sqrt{1 + \sqrt{1 - 2 x}}) = \frac{1}{2} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{- 1}{2}} \frac{d}{d x} {(1 + \sqrt{1 - 2 x})}^{\frac{1}{2}} = \frac{1}{2} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{- 1}{2}} (\frac{1}{2} {(1 + \sqrt{1 - 2 x})}^{- \frac{1}{2}} (\frac{d}{d x} {(1 - 2 x)}^{\frac{1}{2}})) = \frac{1}{2} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{- 1}{2}} (\frac{1}{2} {(1 + \sqrt{1 - 2 x})}^{- \frac{1}{2}} (\frac{1}{2} {(1 - 2 x)}^{- \frac{1}{2}} (- 2))) = \frac{1}{2} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{- 1}{2}} (\frac{1}{2} {(1 + \sqrt{1 - 2 x})}^{- \frac{1}{2}} ({(1 - 2 x)}^{- \frac{1}{2}})) = - \frac{1}{4} {(1 + \sqrt{1 + \sqrt{1 - 2 x}})}^{\frac{- 1}{2}} (\frac{{(1 + \sqrt{1 - 2 x})}^{\frac{- 1}{2}}}{\sqrt{1 - 2 x}}) = - \frac{1}{4 \sqrt{1 + \sqrt{1 + \sqrt{1 - 2 x}}} \sqrt{1 + \sqrt{1 - 2 x}} \sqrt{1 - 2 x}}$

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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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)=1+1+1-2x

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

Mechanics Maths

Decision Maths

Logic and Functions

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Applied Mathematics

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Study anywhere. Anytime. Across all devices.

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