Question

Find the derivative of the function below in two ways. F(x) = x − 5x x x (a) by using the Quotient Rule F'(x) = (b) by simplifying first F'(x) = (c) Which method appears to be simpler for this problem? the Quotient Rule simplifying first

Answer 1

Answer:

(a)  $\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5$

(b)  $\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5$

(c)  Simplifying first

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                          $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                        $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:                                                                            $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Derivative Rule [Quotient Rule]:                                                                          $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle F(x) = \frac{x - 5x\sqrt{x}}{\sqrt{x}}$

Step 2: Differentiate Way 1

1. Derivative Rule [Quotient Rule]:                                                                  $\displaystyle F'(x) = \frac{\sqrt{x} \big( x - 5x\sqrt{x} \big) ' - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}$
2. Rewrite [Derivative Property - Addition/Subtraction]:                               $\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( 5x\sqrt{x} \big) ' \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}$
3. Derivative Rule [Product Rule]:                                                                   $\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( (5x)' \sqrt{x} + 5x(\sqrt{x})' \big) \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}$
4. Rewrite [Derivative Rule - Multiplied Constant]:                                        $\displaystyle F'(x) = \frac{\sqrt{x} \Big[ \big( x \big) '- \big( 5(x)' \sqrt{x} + 5x(\sqrt{x})' \big) \Big] - \big( \sqrt{x} \big) ' \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}$
5. Derivative Rule [Basic Power Rule]:                                                            $\displaystyle F'(x) = \frac{\sqrt{x} \Big[ 1 - \big( 5\sqrt{x} + \frac{5x}{2\sqrt{x}} \big) \Big] - \frac{1}{2\sqrt{x}} \big( x - 5x\sqrt{x} \big) }{\big( \sqrt{x} \big) ^2}$
6. Simplify:                                                                                                        $\displaystyle F'(x) = \frac{\sqrt{x} \Big( 1 - 5\sqrt{x} - \frac{5x}{2\sqrt{x}} \Big) - \frac{x}{2\sqrt{x}} + \frac{5x\sqrt{x}}{2\sqrt{x}}}{x}$
7. Simplify:                                                                                                        $\displaystyle F'(x) = \frac{\sqrt{x} - 5x - \frac{5x}{2} - \frac{x}{2\sqrt{x}} + \frac{5x}{2}}{x}$
8. Simplify:                                                                                                        $\displaystyle F'(x) = \frac{\frac{\sqrt{x}}{2} - 5x}{x}$
9. Simplify:                                                                                                        $\displaystyle F'(x) = \frac{- \Big( 10\sqrt{x} - 1 \Big) }{2\sqrt{x}}$
10. Rewrite:                                                                                                         $\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5$

∴ we find the derivative of the given function but it is a tedious method of computation.

Step 3: Differentiate Way 2

1. [Function] Rewrite:                                                                                       $\displaystyle F(x) = \frac{x}{\sqrt{x}} - \frac{5x\sqrt{x}}{\sqrt{x}}$
2. [Function] Simplify:                                                                                       $\displaystyle F(x) = \sqrt{x} - 5x$
3. [Derivative] Rewrite [Derivative Property - Addition/Subtraction]:           $\displaystyle F'(x) = \big( \sqrt{x} \big) ' - \big( 5x \big) '$
4. Rewrite [Derivative Property - Multiplied Constant]:                                 $\displaystyle F'(x) = \big( \sqrt{x} \big) ' - 5 \big( x \big) '$
5. Derivative Rule [Basic Power Rule]:                                                            $\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5$

∴ we find the derivative of the given function and it is less complex and faster. We can conclude that simplifying first appears to be simpler for this problem.

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation