Example: Finding the rate of change of \( \ln(\sqrt{x}) \)

Step 1: Start with the Original Expression

We have \( \ln(\sqrt{x}) \) and we want to find its rate of change.

Step 2: Rewrite Using Exponents

First, change \( \sqrt{x} \) to \( x^{\frac{1}{2}} \). So, the expression becomes \( \ln(x^{\frac{1}{2}}) \).

Step 3: Use Logarithmic Properties to Simplify

Bring down the exponent \( \frac{1}{2} \) to the front. Now, the expression is \( \frac{1}{2} \ln(x) \).

Step 4: Find the Rate of Change

We want to find how \( \frac{1}{2} \ln(x) \) changes as \( x \) changes. The rate of change of \( \ln(x) \) is \( \frac{1}{x} \).

Step 5: Multiply by \( \frac{1}{2} \)

Finally, we have \( \frac{1}{2} \) times \( \frac{1}{x} \), which simplifies to \( \frac{1}{2x} \).

Step 6: Final Answer

The rate of change of \( \ln(\sqrt{x}) \) is \( \frac{1}{2x} \).

Step 1: Start with the Original Expression We have ln(√x) and we want to find its rate of change. Step 2: Rewrite Using Exponents First, change √x to x⁰.⁵. So, the expression becomes ln(x⁰.⁵). Step 3: Use Logarithmic Properties to Simplify Bring down the exponent 0.5 to the front. Now, the expression is 0.5 ln(x). Step 4: Find the Rate of Change We want to find how 0.5 ln(x) changes as x changes. The rate of change of ln(x) is 1/x. Step 5: Multiply by 0.5 Finally, we have 0.5 times 1/x, which simplifies to 0.5/x. Step 6: Final Answer The rate of change of ln(√x) is 0.5/x. I hope this breakdown in Unicode math formatting is more accessible and in line with your preferences.