Estimate the instantaneous rate of change of P(x)=5/x+2 at the point x=1. The answer should be accurate to at least 3 decimal places.

To estimate the instantaneous rate of change of the function \( P(x) = \frac{5}{x} + 2 \) at the point \( x = 1 \), we need to calculate the derivative of \( P(x) \) and then evaluate it at \( x = 1 \).

### 1. Find the Derivative
First, let's find the derivative of \( P(x) \).

\[
P(x) = \frac{5}{x} + 2
\]

The derivative of \( P(x) \) with respect to \( x \) is:

\[
P'(x) = \frac{d}{dx}\left(\frac{5}{x} + 2\right)
\]

The derivative of \( \frac{5}{x} \) is \( -\frac{5}{x^2} \) and the derivative of the constant \( 2 \) is 0, so:

\[
P'(x) = -\frac{5}{x^2}
\]

### 2. Evaluate the Derivative at \( x = 1 \)
Now, substitute \( x = 1 \) into \( P'(x) \) to find the instantaneous rate of change at \( x = 1 \):

\[
P'(1) = -\frac{5}{1^2} = -5
\]

### 3. Conclusion
The instantaneous rate of change of \( P(x) = \frac{5}{x} + 2 \) at the point \( x = 1 \) is \( \boxed{-5} \). This answer is accurate to three decimal places.

Would you like further details or explanations?

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Here are five related questions:
1. What is the physical meaning of the instantaneous rate of change?
2. How can you find the instantaneous rate of change at points where the function is not differentiable?
3. What are the general rules for finding derivatives of functions of the form \( \frac{a}{x} \)?
4. How does the rate of change of \( P(x) \) behave as \( x \) approaches 0?
5. How would you estimate the instantaneous rate of change using a difference quotient instead of calculus?

**Tip:** When calculating derivatives of functions like \( \frac{a}{x} \), remember that the power rule can be applied if you rewrite the function as \( ax^{-1} \).

Learn how to estimate the instantaneous rate of change of the function P(x) = 5/x + 2 at the point x = 1 with a detailed step-by-step solution. This problem covers calculus concepts, including derivatives and the formula for instantaneous rate of change. Suitable for advanced high school or college students.

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