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Chapter 2: Q. 52 (page 184)

Use the definition of the derivative to find $f$ for each function $f$ in Exercises $39 - 54$
$f (x) = \frac{1}{x^{2} - 1}$

Short Answer

Expert verified

The derivative is $f^{'} (x) = - \frac{1}{{(x^{2} - 1)}^{2}}$

Step by step solution

Given information

Given function $f (x) = \frac{1}{x^{2} - 1}$

Use the rule to calculate the derivative

Calculating, we get

$f (x) = \frac{1}{x^{2} - 1} f' (x) = {(x^{2} - 1)}^{- 1} f' (x) = - {(x^{2} - 1)}^{- 1 - 1} f' (x) = - {(x^{2} - 1)}^{- 2} f' (x) = - \frac{1}{{(x^{2} - 1)}^{2}}$

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Use the definition of the derivative to prove the following special case of the product rule

$\frac{d}{d x} (x^{2} f (x)) = 2 x f (x) + x^{2} f' (x)$

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:

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Use (a) the $h \to 0$ definition of the derivative and then

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

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Use the definition of the derivative to find ffor each function fin Exercises 39-54f(x)=1x2-1

Short Answer

Step by step solution

Given information

Use the rule to calculate the derivative

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Use the definition of the derivative to find $f$ for each function $f$ in Exercises $39 - 54$
$f (x) = \frac{1}{x^{2} - 1}$