Questions: For the function, find the following. f(x) = 8 + 9x + 1/2 x^2 + 5/6 x^3 + 5/24 x^4 + 7/120 x^5 (a) f'(x) (b) f''(x) (c) f'''(x) (d) f^(4)(x)

For the function, find the following. f(x) = 8 + 9x + 1/2 x^2 + 5/6 x^3 + 5/24 x^4 + 7/120 x^5 (a) f'(x) (b) f''(x) (c) f'''(x) (d) f^(4)(x)
Transcript text: For the function, find the following. \[ f(x)=8+9 x+\frac{1}{2} x^{2}+\frac{5}{6} x^{3}+\frac{5}{24} x^{4}+\frac{7}{120} x^{5} \] (a) $f^{\prime}(x)$ (b) $f^{\prime \prime}(x)$ (c) $f^{\prime \prime \prime}(x)$ (d) $f^{(4)}(x)$
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Solution

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Solution Steps

Step 1: Define the Function

The function is defined as: \[ f(x) = 8 + 9x + \frac{1}{2}x^{2} + \frac{5}{6}x^{3} + \frac{5}{24}x^{4} + \frac{7}{120}x^{5} \]

Step 2: Calculate the First Derivative

The first derivative \( f'(x) \) is calculated using the power rule: \[ f'(x) = \frac{d}{dx}\left(8 + 9x + \frac{1}{2}x^{2} + \frac{5}{6}x^{3} + \frac{5}{24}x^{4} + \frac{7}{120}x^{5}\right) = \frac{29}{120}x^{4} + \frac{5}{6}x^{3} + x^{2} + 9 \]

Step 3: Calculate the Second Derivative

The second derivative \( f''(x) \) is obtained by differentiating \( f'(x) \): \[ f''(x) = \frac{d}{dx}\left(\frac{29}{120}x^{4} + \frac{5}{6}x^{3} + x^{2} + 9\right) = \frac{29}{30}x^{3} + \frac{5}{2}x^{2} + 5 \]

Step 4: Calculate the Third Derivative

The third derivative \( f'''(x) \) is found by differentiating \( f''(x) \): \[ f'''(x) = \frac{d}{dx}\left(\frac{29}{30}x^{3} + \frac{5}{2}x^{2} + 5\right) = \frac{29}{10}x^{2} + 5x + 5 \]

Final Answer

(a) \( \boxed{f'(x) = \frac{29}{120}x^{4} + \frac{5}{6}x^{3} + x^{2} + 9} \)
(b) \( \boxed{f''(x) = \frac{29}{30}x^{3} + \frac{5}{2}x^{2} + 5} \)
(c) \( \boxed{f'''(x) = \frac{29}{10}x^{2} + 5x + 5} \)
(d) \( \boxed{f^{(4)}(x) = \frac{29}{5}x + 5} \)

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