As a mixed number, 16 1/4, as a fraction, 65/4.
Answer:
(a) ![\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%205)
(b) ![\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%205)
(c) Simplifying first
General Formulas and Concepts:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: ![\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bcf%28x%29%5D%20%3D%20c%20%5Ccdot%20f%27%28x%29)
Derivative Property [Addition/Subtraction]: ![\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%20%2B%20g%28x%29%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D)
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- 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)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bf%28x%29g%28x%29%5D%3Df%27%28x%29g%28x%29%20%2B%20g%27%28x%29f%28x%29)
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)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify.</em>
<em />![\displaystyle F(x) = \frac{x - 5x\sqrt{x}}{\sqrt{x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%28x%29%20%3D%20%5Cfrac%7Bx%20-%205x%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%7D%7D)
<u>Step 2: Differentiate Way 1</u>
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20-%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%7D%7B%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%5E2%7D)
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5CBig%5B%20%5Cbig%28%20x%20%5Cbig%29%20%27-%20%5Cbig%28%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20%5CBig%5D%20-%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%7D%7B%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%5E2%7D)
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5CBig%5B%20%5Cbig%28%20x%20%5Cbig%29%20%27-%20%5Cbig%28%20%285x%29%27%20%5Csqrt%7Bx%7D%20%2B%205x%28%5Csqrt%7Bx%7D%29%27%20%5Cbig%29%20%5CBig%5D%20-%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%7D%7B%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%5E2%7D)
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5CBig%5B%20%5Cbig%28%20x%20%5Cbig%29%20%27-%20%5Cbig%28%205%28x%29%27%20%5Csqrt%7Bx%7D%20%2B%205x%28%5Csqrt%7Bx%7D%29%27%20%5Cbig%29%20%5CBig%5D%20-%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%7D%7B%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%5E2%7D)
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5CBig%5B%201%20-%20%5Cbig%28%205%5Csqrt%7Bx%7D%20%2B%20%5Cfrac%7B5x%7D%7B2%5Csqrt%7Bx%7D%7D%20%5Cbig%29%20%5CBig%5D%20-%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20%5Cbig%28%20x%20-%205x%5Csqrt%7Bx%7D%20%5Cbig%29%20%7D%7B%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%5E2%7D)
- 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}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20%5CBig%28%201%20-%205%5Csqrt%7Bx%7D%20-%20%5Cfrac%7B5x%7D%7B2%5Csqrt%7Bx%7D%7D%20%5CBig%29%20-%20%5Cfrac%7Bx%7D%7B2%5Csqrt%7Bx%7D%7D%20%2B%20%5Cfrac%7B5x%5Csqrt%7Bx%7D%7D%7B2%5Csqrt%7Bx%7D%7D%7D%7Bx%7D)
- Simplify:
![\displaystyle F'(x) = \frac{\sqrt{x} - 5x - \frac{5x}{2} - \frac{x}{2\sqrt{x}} + \frac{5x}{2}}{x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%20-%205x%20-%20%5Cfrac%7B5x%7D%7B2%7D%20-%20%5Cfrac%7Bx%7D%7B2%5Csqrt%7Bx%7D%7D%20%2B%20%5Cfrac%7B5x%7D%7B2%7D%7D%7Bx%7D)
- Simplify:
![\displaystyle F'(x) = \frac{\frac{\sqrt{x}}{2} - 5x}{x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B%5Cfrac%7B%5Csqrt%7Bx%7D%7D%7B2%7D%20-%205x%7D%7Bx%7D)
- Simplify:
![\displaystyle F'(x) = \frac{- \Big( 10\sqrt{x} - 1 \Big) }{2\sqrt{x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B-%20%5CBig%28%2010%5Csqrt%7Bx%7D%20-%201%20%5CBig%29%20%7D%7B2%5Csqrt%7Bx%7D%7D)
- Rewrite:
![\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%205)
∴ we find the derivative of the given function but it is a tedious method of computation.
<u>Step 3: Differentiate Way 2</u>
- [Function] Rewrite:
![\displaystyle F(x) = \frac{x}{\sqrt{x}} - \frac{5x\sqrt{x}}{\sqrt{x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%28x%29%20%3D%20%5Cfrac%7Bx%7D%7B%5Csqrt%7Bx%7D%7D%20-%20%5Cfrac%7B5x%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%7D%7D)
- [Function] Simplify:
![\displaystyle F(x) = \sqrt{x} - 5x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%28x%29%20%3D%20%5Csqrt%7Bx%7D%20-%205x)
- [Derivative] Rewrite [Derivative Property - Addition/Subtraction]:
![\displaystyle F'(x) = \big( \sqrt{x} \big) ' - \big( 5x \big) '](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20-%20%5Cbig%28%205x%20%5Cbig%29%20%27)
- Rewrite [Derivative Property - Multiplied Constant]:
![\displaystyle F'(x) = \big( \sqrt{x} \big) ' - 5 \big( x \big) '](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cbig%28%20%5Csqrt%7Bx%7D%20%5Cbig%29%20%27%20-%205%20%5Cbig%28%20x%20%5Cbig%29%20%27)
- Derivative Rule [Basic Power Rule]:
![\displaystyle F'(x) = \frac{1}{2\sqrt{x}} - 5](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%205)
∴ we find the derivative of the given function <em>and </em>it is less complex and faster. We can conclude that simplifying first appears to be simpler for this problem.
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Learn more about differentiation: brainly.com/question/17830594
Learn more about calculus: brainly.com/question/23558817
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation