List of six derivative functions with complex algebraic expressions.
A mathematical image displaying six derivative functions, each labeled from 1 to 6, involving algebraic expressions with square roots, fractions, and polynomial terms.
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Step-by-step solution for: Find Derivatives of Functions in Calculus
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Show Answer Key & Explanations
Step-by-step solution for: Find Derivatives of Functions in Calculus
It seems you've uploaded a list of derivative functions, and the task is to solve or analyze these derivatives. However, since no specific problem or question has been stated, I will assume the task is to verify or simplify each derivative expression. Let's go through each one step by step.
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This expression is already in a simplified form. It represents the derivative of some function \( f(x) \). There is no further simplification needed unless additional context is provided.
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This expression is also in a simplified form. It represents the derivative of a function involving a rational term. No further simplification is possible without additional context.
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We can simplify this expression by dividing each term in the numerator by \( 2x^2 \):
\[
f'(x) = \frac{12x^{5/2}}{2x^2} - \frac{8x^2}{2x^2} + \frac{5x^{3/2}}{2x^2} - \frac{\sqrt{x}}{2x^2} + \frac{2}{2x^2}
\]
Simplify each term:
1. \( \frac{12x^{5/2}}{2x^2} = \frac{12}{2} \cdot x^{5/2 - 2} = 6x^{1/2} = 6\sqrt{x} \)
2. \( \frac{8x^2}{2x^2} = \frac{8}{2} = 4 \)
3. \( \frac{5x^{3/2}}{2x^2} = \frac{5}{2} \cdot x^{3/2 - 2} = \frac{5}{2}x^{-1/2} = \frac{5}{2\sqrt{x}} \)
4. \( \frac{\sqrt{x}}{2x^2} = \frac{x^{1/2}}{2x^2} = \frac{1}{2}x^{1/2 - 2} = \frac{1}{2}x^{-3/2} = \frac{1}{2x^{3/2}} \)
5. \( \frac{2}{2x^2} = \frac{1}{x^2} \)
Putting it all together:
\[
f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2}
\]
This is the simplified form.
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This expression is already in a compact form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
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This expression is also in a simplified form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
---
This expression is already in a simplified form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
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1. \( f'(x) = \frac{\sqrt{x}(7x^2 + 6)}{2} \)
2. \( f'(x) = \frac{-3}{(4x+7)^2} \)
3. \( f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2} \)
4. \( f'(x) = \frac{-4x^{5/2} + 3x^2 + 4\sqrt{x} + 1}{2\sqrt{x}(x^2 - 1)^2} \)
5. \( f'(x) = \frac{4x^3 + 1}{\sqrt{2x^4 + 2x - 1}} \)
6. \( f'(x) = \frac{9x^2 - 4x + 9}{2(x^2 - 3)^2 \sqrt{3x - 1}} \)
If there is a specific task related to these derivatives (e.g., finding critical points, analyzing behavior, etc.), please provide additional details so I can assist further.
\[
\boxed{
\begin{aligned}
1. & \quad f'(x) = \frac{\sqrt{x}(7x^2 + 6)}{2} \\
2. & \quad f'(x) = \frac{-3}{(4x+7)^2} \\
3. & \quad f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2} \\
4. & \quad f'(x) = \frac{-4x^{5/2} + 3x^2 + 4\sqrt{x} + 1}{2\sqrt{x}(x^2 - 1)^2} \\
5. & \quad f'(x) = \frac{4x^3 + 1}{\sqrt{2x^4 + 2x - 1}} \\
6. & \quad f'(x) = \frac{9x^2 - 4x + 9}{2(x^2 - 3)^2 \sqrt{3x - 1}}
\end{aligned}
}
\]
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1. \( f'(x) = \frac{\sqrt{x}(7x^2 + 6)}{2} \)
This expression is already in a simplified form. It represents the derivative of some function \( f(x) \). There is no further simplification needed unless additional context is provided.
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2. \( f'(x) = \frac{-3}{(4x+7)^2} \)
This expression is also in a simplified form. It represents the derivative of a function involving a rational term. No further simplification is possible without additional context.
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3. \( f'(x) = \frac{12x^{5/2} - 8x^2 + 5x^{3/2} - \sqrt{x} + 2}{2x^2} \)
We can simplify this expression by dividing each term in the numerator by \( 2x^2 \):
\[
f'(x) = \frac{12x^{5/2}}{2x^2} - \frac{8x^2}{2x^2} + \frac{5x^{3/2}}{2x^2} - \frac{\sqrt{x}}{2x^2} + \frac{2}{2x^2}
\]
Simplify each term:
1. \( \frac{12x^{5/2}}{2x^2} = \frac{12}{2} \cdot x^{5/2 - 2} = 6x^{1/2} = 6\sqrt{x} \)
2. \( \frac{8x^2}{2x^2} = \frac{8}{2} = 4 \)
3. \( \frac{5x^{3/2}}{2x^2} = \frac{5}{2} \cdot x^{3/2 - 2} = \frac{5}{2}x^{-1/2} = \frac{5}{2\sqrt{x}} \)
4. \( \frac{\sqrt{x}}{2x^2} = \frac{x^{1/2}}{2x^2} = \frac{1}{2}x^{1/2 - 2} = \frac{1}{2}x^{-3/2} = \frac{1}{2x^{3/2}} \)
5. \( \frac{2}{2x^2} = \frac{1}{x^2} \)
Putting it all together:
\[
f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2}
\]
This is the simplified form.
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4. \( f'(x) = \frac{-4x^{5/2} + 3x^2 + 4\sqrt{x} + 1}{2\sqrt{x}(x^2 - 1)^2} \)
This expression is already in a compact form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
---
5. \( f'(x) = \frac{4x^3 + 1}{\sqrt{2x^4 + 2x - 1}} \)
This expression is also in a simplified form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
---
6. \( f'(x) = \frac{9x^2 - 4x + 9}{2(x^2 - 3)^2 \sqrt{3x - 1}} \)
This expression is already in a simplified form. The numerator and denominator are both well-defined, and no further simplification is immediately apparent without additional context.
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Summary of Simplified Forms
1. \( f'(x) = \frac{\sqrt{x}(7x^2 + 6)}{2} \)
2. \( f'(x) = \frac{-3}{(4x+7)^2} \)
3. \( f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2} \)
4. \( f'(x) = \frac{-4x^{5/2} + 3x^2 + 4\sqrt{x} + 1}{2\sqrt{x}(x^2 - 1)^2} \)
5. \( f'(x) = \frac{4x^3 + 1}{\sqrt{2x^4 + 2x - 1}} \)
6. \( f'(x) = \frac{9x^2 - 4x + 9}{2(x^2 - 3)^2 \sqrt{3x - 1}} \)
If there is a specific task related to these derivatives (e.g., finding critical points, analyzing behavior, etc.), please provide additional details so I can assist further.
Final Answer:
\[
\boxed{
\begin{aligned}
1. & \quad f'(x) = \frac{\sqrt{x}(7x^2 + 6)}{2} \\
2. & \quad f'(x) = \frac{-3}{(4x+7)^2} \\
3. & \quad f'(x) = 6\sqrt{x} - 4 + \frac{5}{2\sqrt{x}} - \frac{1}{2x^{3/2}} + \frac{1}{x^2} \\
4. & \quad f'(x) = \frac{-4x^{5/2} + 3x^2 + 4\sqrt{x} + 1}{2\sqrt{x}(x^2 - 1)^2} \\
5. & \quad f'(x) = \frac{4x^3 + 1}{\sqrt{2x^4 + 2x - 1}} \\
6. & \quad f'(x) = \frac{9x^2 - 4x + 9}{2(x^2 - 3)^2 \sqrt{3x - 1}}
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of calculus answers.
