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Chapter 2: Q 9. (page 237)

Find the derivatives of the function: ${(x \sqrt{2 x - 1})}^{- 3} .$

Short Answer

Expert verified

The derivative of ${(x \sqrt{2 x - 1})}^{- 3} i s - 3 {(x \sqrt{2 x - 1})}^{- 4} . \frac{(3 x - 1)}{\sqrt{2 x - 1}} .$

Step by step solution

01

Given information

The given expression is ${(x \sqrt{2 x - 1})}^{- 3} .$

02

Simplification

$f' (x) = \frac{d}{d x} (f (x)) = \frac{d}{d x} {{(x \sqrt{2 x - 1})}^{- 3}} = - 3 {(x \sqrt{2 x - 1})}^{- 4} . \frac{d}{d x} (x \sqrt{2 x - 1}) = - 3 {(x \sqrt{2 x - 1})}^{- 4} . (1 . \sqrt{2 x - 1} + x . \frac{2}{2 \sqrt{2 x - 1}}) = - 3 {(x \sqrt{2 x - 1})}^{- 4} . (\frac{2 x - 1 + x}{\sqrt{2 x - 1}}) = - 3 {(x \sqrt{2 x - 1})}^{- 4} . \frac{(3 x - 1)}{\sqrt{2 x - 1}} H e n c e, f' (x) = - 3 {(x \sqrt{2 x - 1})}^{- 4} . \frac{(3 x - 1)}{\sqrt{2 x - 1}} .$

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Most popular questions from this chapter

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.

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

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54

role="math" localid="1648290170541" $f (x) = \frac{x - 1}{x + 3}$

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.

$f (x) = \sqrt{3 x - 4 {(2 x + 1)}^{6}}$

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

$f' (x) = 1 - 4 x^{6}; f (1) = 3$

For each function $f$ graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

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