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Chapter 2: Q. 53 (page 198)

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.
$f (x) = \frac{\sqrt[5]{x^{7}} - 2 x^{4}}{x^{3}}$

Short Answer

Expert verified

The derivative of the function is $\begin{array}{rcl} \frac{- 8}{5} x^{- \frac{13}{5}} - 2 \end{array}$ .

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \frac{\sqrt[5]{x^{7}} - 2 x^{4}}{x^{3}}$ .

02

Step 2. Simplify the function

Simplify the function.

$\begin{array}{rcl} f (x) & = & \frac{\sqrt[5]{x^{7}}}{x^{3}} - \frac{2 x^{4}}{x^{3}} \\ = & \frac{x^{\frac{7}{5}}}{x^{3}} - 2 x \\ = & x^{\frac{7}{5} - 3} - 2 x \\ = & x^{- \frac{8}{5}} - 2 x \end{array}$

03

Step 3. Find the derivative

Apply the difference rule of derivative, $(f - g)' (x) = f' (x) - g' (x)$ .

$f' (x) = \frac{d}{d x} x^{- \frac{8}{5}} - \frac{d}{d x} (2 x)$

Apply constant multiple of derivative, $(k f)' (x) = k f' (x)$ .

localid="1648555315954" $f' (x) = \frac{d}{d x} x^{\frac{- 8}{5}} - 2 \frac{d}{d x} (x)$

Apply power rule of derivative, $(x^{n})' = n x^{n - 1}$ .

localid="1648555489882" $\begin{array}{rcl} f' (x) & = & \frac{- 8}{5} (x^{\frac{- 8}{5} - 1}) - 2 (1) \\ = & \frac{- 8}{5} x^{- \frac{13}{5}} - 2 \end{array}$

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Most popular questions from this chapter

use the definition of derivative to directly prove the differentiation rules for constant and identity function

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.

$f (x) = x^{- \frac{1}{2}} {(x^{2} - 1)}^{3}$

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

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

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.

23. $f (x) = x^{2}, x = - 3$

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

The tangent line to $f (x) = x^{2}$ at $x = - 3$

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