Consider the following.

*y* = ln(*x*5),    (1, 0)

(a)

Find *y* ′.

To find \( y' \) (the derivative of \( y \) with respect to \( x \)), given the function \( y = \ln(x^5) \) and the point \( (1, 0) \), we'll proceed as follows:

### Step 1: Simplify the logarithmic expression
We can use the logarithm power rule:
\[
y = \ln(x^5) = 5\ln(x)
\]

### Step 2: Differentiate the function
Now, differentiate \( y = 5\ln(x) \) with respect to \( x \):
\[
y' = \frac{d}{dx}[5\ln(x)] = 5 \cdot \frac{d}{dx}[\ln(x)]
\]
\[
y' = 5 \cdot \frac{1}{x} = \frac{5}{x}
\]

### Step 3: Evaluate the derivative at the given point
We are asked to find the derivative at the point \( (1, 0) \). Since \( x = 1 \) at this point:
\[
y' = \frac{5}{1} = 5
\]

So, the derivative \( y' \) at the point \( (1, 0) \) is:
\[
\boxed{5}
\]

Would you like to explore any more details or have additional questions? Here are some related questions to consider:

1. How does the power rule for logarithms work?
2. What is the derivative of a natural logarithm function?
3. How do you interpret the derivative of a logarithmic function?
4. How do you apply the derivative at a specific point?
5. What are the properties of logarithmic functions?

**Tip:** When differentiating functions involving logarithms, it's often helpful to simplify the logarithmic expression using the logarithm rules before differentiating. This can make the derivative much easier to compute.

Learn how to find the derivative of ln(x^5) using the logarithm power rule and evaluate it at the point (1, 0). This problem involves logarithmic functions and derivatives, suitable for advanced high school students.

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