Questions: 19. f(x) = 9/x^5 + 5 sqrt(x)/4 - 11 20. f(x) = 2 sqrt(x^5) + 1/(2 sqrt(x^5)) 21. f(x) = (3 x^4 + 5 x^2 - 2 x)/(4 x^2)

19. f(x) = 9/x^5 + 5 sqrt(x)/4 - 11 20. f(x) = 2 sqrt(x^5) + 1/(2 sqrt(x^5)) 21. f(x) = (3 x^4 + 5 x^2 - 2 x)/(4 x^2)
Transcript text: 19. $f(x)=\frac{9}{x^{5}}+\frac{5 \sqrt{x}}{4}-11$ 20. $f(x)=2 \sqrt{x^{5}}+\frac{1}{2 \sqrt{x^{5}}}$ 21. $f(x)=\frac{3 x^{4}+5 x^{2}-2 x}{4 x^{2}}$
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Solution

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Solution Steps

To find the derivative \( f'(x) \) for each function, we will use the rules of differentiation, including the power rule, the chain rule, and the quotient rule as necessary.

19. \( f(x) = \frac{9}{x^5} + \frac{5 \sqrt{x}}{4} - 11 \)
  • Rewrite the terms to make differentiation easier.
  • Apply the power rule to each term.
20. \( f(x) = 2 \sqrt{x^5} + \frac{1}{2 \sqrt{x^5}} \)
  • Rewrite the terms using exponents.
  • Apply the power rule to each term.
21. \( f(x) = \frac{3x^4 + 5x^2 - 2x}{4x^2} \)
  • Simplify the function if possible.
  • Apply the quotient rule or simplify and then differentiate.
Step 1: Differentiate \( f_1(x) \)

Given the function: \[ f_1(x) = \frac{9}{x^5} + \frac{5 \sqrt{x}}{4} - 11 \] We differentiate \( f_1(x) \): \[ f_1'(x) = -\frac{45}{x^6} + \frac{5}{8\sqrt{x}} \]

Step 2: Differentiate \( f_2(x) \)

Given the function: \[ f_2(x) = 2 \sqrt{x^5} + \frac{1}{2 \sqrt{x^5}} \] We differentiate \( f_2(x) \): \[ f_2'(x) = \frac{5\sqrt{x^5}}{x} - \frac{5}{4x\sqrt{x^5}} \]

Step 3: Differentiate \( f_3(x) \)

Given the function: \[ f_3(x) = \frac{3x^4 + 5x^2 - 2x}{4x^2} \] We differentiate \( f_3(x) \): \[ f_3'(x) = \frac{12x^3 + 10x - 2}{4x^2} - \frac{3x^4 + 5x^2 - 2x}{2x^3} \]

Final Answer

The derivatives are: \[ f_1'(x) = -\frac{45}{x^6} + \frac{5}{8\sqrt{x}}, \quad f_2'(x) = \frac{5\sqrt{x^5}}{x} - \frac{5}{4x\sqrt{x^5}}, \quad f_3'(x) = \frac{12x^3 + 10x - 2}{4x^2} - \frac{3x^4 + 5x^2 - 2x}{2x^3} \] Thus, the final boxed answers are: \[ \boxed{f_1'(x) = -\frac{45}{x^6} + \frac{5}{8\sqrt{x}}} \] \[ \boxed{f_2'(x) = \frac{5\sqrt{x^5}}{x} - \frac{5}{4x\sqrt{x^5}}} \] \[ \boxed{f_3'(x) = \frac{12x^3 + 10x - 2}{4x^2} - \frac{3x^4 + 5x^2 - 2x}{2x^3}} \]

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