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Let's calculate the first and second derivatives of the function \( f(x) = \frac{5x + 3}{x} \).

First, simplify the function:
\[
f(x) = \frac{5x + 3}{x} = 5 + \frac{3}{x}
\]

Now, we'll find the first derivative, \( f'(x) \), using the power rule and the rule for differentiating \( \frac{1}{x} \):
\[
f'(x) = 0 - \frac{3}{x^2} = -\frac{3}{x^2}
\]

Next, we calculate the second derivative, \( f''(x) \), by differentiating \( f'(x) \) again:
\[
f''(x) = \frac{6}{x^3}
\]

Thus, the first and second derivatives of the function are:
\[
f'(x) = -\frac{3}{x^2}, \quad f''(x) = \frac{6}{x^3}
\]

Would you like further details on these calculations or have any other questions?

Related questions:
1. How would you integrate \( f(x) = 5 + \frac{3}{x} \)?
2. What happens to \( f'(x) \) and \( f''(x) \) as \( x \) approaches infinity?
3. How would the graph of \( f(x) \), \( f'(x) \), and \( f''(x) \) look?
4. Can you solve \( f'(x) = 0 \) to find the critical points of \( f(x) \)?
5. How does knowing the derivatives help in understanding the behavior of the function?

**Tip:** When working with rational functions, always consider simplifying the function first to make differentiation easier.

Find f'(x) and f''(x) for the function f(x) = (5x + 3) / x.

Learn how to find the first and second derivatives of the function f(x) = (5x + 3) / x. This detailed solution covers key calculus concepts such as the power rule and quotient rule, making it suitable for high school students in Grades 11-12.

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