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8. Consider the following.
y = ln(x5), (1, 0)
8 Find the slope of the line tangent to the graph of y at the given point
Find an equation of the tangent line to the graph of the function at the given point.


To solve the problem given:

### Problem Statement:
1. **Given function:** \( y = \ln(x^5) \)
2. **Point of tangency:** \( (1, 0) \)

### Steps to Solve:
1. **Find the derivative \( \frac{dy}{dx} \)** to determine the slope of the tangent line.
2. **Evaluate the derivative at the point (1, 0)** to get the slope at that specific point.
3. **Use the point-slope form of a line** to find the equation of the tangent line.

### 1. Derivative of \( y = \ln(x^5) \):

The function can be simplified using the logarithmic property:

\[
y = \ln(x^5) = 5\ln(x)
\]

Now, differentiate \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{d}{dx}[5\ln(x)] = \frac{5}{x}
\]

### 2. Evaluate the derivative at \( x = 1 \):

\[
\text{Slope} = \frac{5}{1} = 5
\]

### 3. Equation of the Tangent Line:

Using the point-slope form of the equation:

\[
y - y_1 = m(x - x_1)
\]

Where \( m = 5 \), \( x_1 = 1 \), and \( y_1 = 0 \), the equation becomes:

\[
y - 0 = 5(x - 1)
\]

Simplifying this:

\[
y = 5x - 5
\]

### Final Answers:
1. **Slope of the tangent line:** \( 5 \)
2. **Equation of the tangent line:** \( y = 5x - 5 \)

Would you like more details on any step or have any questions?

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Here are some related questions you might find interesting:

1. How do you find the second derivative of \( y = \ln(x^5) \) and what does it represent?
2. What is the geometric interpretation of the slope of a tangent line?
3. How does the natural logarithm function behave for different values of \( x \)?
4. How can the chain rule be applied to more complex logarithmic functions?
5. What is the relationship between the derivative and the concavity of a function?

**Tip:** When working with logarithmic functions, always consider simplifying the expression using logarithmic identities before differentiating. This can make the differentiation process easier and clearer.

Learn how to find the slope of the tangent line and its equation for the function y = ln(x^5) at the point (1, 0). This problem involves differentiation of logarithmic functions and application of the point-slope form of a line. Suitable for advanced high school students and calculus enthusiasts.

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