the slope of the tangent line to the curve y=3/x at the point (8,3/8) is

To find the slope of the tangent line to the curve \( y = \frac{3}{x} \) at the point \( (8, \frac{3}{8}) \), we need to compute the derivative of \( y = \frac{3}{x} \) with respect to \( x \), which represents the slope of the tangent line at any point on the curve.

1. **Find the derivative of \( y \) with respect to \( x \):**

\[
y = \frac{3}{x} \quad \Rightarrow \quad \frac{dy}{dx} = -\frac{3}{x^2}
\]

2. **Evaluate the derivative at \( x = 8 \):**

\[
\frac{dy}{dx} \bigg|_{x=8} = -\frac{3}{8^2} = -\frac{3}{64}
\]

So, the slope of the tangent line to the curve \( y = \frac{3}{x} \) at the point \( (8, \frac{3}{8}) \) is \( -\frac{3}{64} \).

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How is the derivative related to the slope of the tangent line?
2. What does the negative slope of the tangent line indicate about the curve at \( x = 8 \)?
3. Can you find the equation of the tangent line at \( (8, \frac{3}{8}) \)?
4. How would the slope change if the function was \( y = \frac{5}{x} \)?
5. What is the geometric interpretation of the derivative at a point?

**Tip**: Always remember that the derivative of a function at a point gives the instantaneous rate of change or the slope of the tangent line at that point.

Discover how to find the slope of the tangent line to the curve y = 3/x at the point (8, 3/8). This detailed solution uses derivative calculus, covering key concepts like finding the slope of a curve at a point. Ideal for students in Grades 11-12 studying advanced calculus.

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