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Chapter 2: Q. 39 (page 210)

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.
$f (x) = 3 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{2}{3}}$

Short Answer

Expert verified

The required answer is $- 2 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{5}{3}} (16 x {(x^{2} + 1)}^{7} - 7)$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = 3 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{2}{3}}$

02

Step 2. Calculation

Differentiate both the sides with respect tox, we get,

$f' (x) = \frac{d}{d x} 3 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{2}{3}} = 3 (- \frac{2}{3}) {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{2}{3} - 1} \frac{d}{d x} ({(x^{2} + 1)}^{8} - 7 x) = - 2 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{5}{3}} (8 {(x^{2} + 1)}^{8 - 1} (2 x) - 7) = - 2 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{5}{3}} (8 {(x^{2} + 1)}^{7} (2 x) - 7) = - 2 {({(x^{2} + 1)}^{8} - 7 x)}^{- \frac{5}{3}} (16 x {(x^{2} + 1)}^{7} - 7)$

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Most popular questions from this chapter

Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .

For each function $f (x)$ and interval localid="1648297458718" $[a, b]$ in Exercises localid="1648297462718" $81 - 86$ , use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" $[a, b]$ . Then apply Newton鈥檚 method to approximate that root.

localid="1648297471865" $f (x) = x^{3} - 3 x + 1, [a, b] = [1, 2]$

A bowling ball dropped from a height of $400$ feet will be $s (t) = 400 - 16 t^{2}$ feet from the ground after $t$ seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After $t = 2$ seconds, with $h = 0.1, h = 0.01 h = - 0.1 a n d h = - 0.01$

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.

$f (x) = \frac{(x + 1) (3 x - 4)}{\sqrt{x^{3} - 27}}$

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.

29. $f (x) = x^{\frac{1}{2}}, x = 9$

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