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We are tasked with finding the derivative of the function \( y = \sqrt{\frac{t}{t+1}} \) using **logarithmic differentiation**. Let's break this down step by step:

### Step 1: Rewrite the equation
We begin by expressing \( y \) in a more convenient form for differentiation:
\[
y = \left( \frac{t}{t+1} \right)^{1/2}
\]
This allows us to work more easily with logarithms.

### Step 2: Take the natural logarithm of both sides
Using logarithmic differentiation, we apply the natural logarithm to both sides of the equation:
\[
\ln y = \ln \left( \left( \frac{t}{t+1} \right)^{1/2} \right)
\]
By the power rule for logarithms, we bring the exponent \( \frac{1}{2} \) in front:
\[
\ln y = \frac{1}{2} \ln \left( \frac{t}{t+1} \right)
\]

### Step 3: Apply the logarithmic property for division
Using the property \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), we get:
\[
\ln y = \frac{1}{2} \left( \ln t - \ln (t+1) \right)
\]

### Step 4: Differentiate both sides with respect to \( t \)
We now differentiate both sides of the equation with respect to \( t \). On the left-hand side, we apply the chain rule:
\[
\frac{1}{y} \frac{dy}{dt} = \frac{1}{2} \left( \frac{1}{t} - \frac{1}{t+1} \right)
\]
The right-hand side is simplified as:
\[
\frac{1}{2} \left( \frac{1}{t} - \frac{1}{t+1} \right) = \frac{1}{2} \cdot \frac{(t+1) - t}{t(t+1)} = \frac{1}{2} \cdot \frac{1}{t(t+1)}
\]

### Step 5: Solve for \( \frac{dy}{dt} \)
Now, multiply both sides by \( y \) to solve for \( \frac{dy}{dt} \):
\[
\frac{dy}{dt} = y \cdot \frac{1}{2} \cdot \frac{1}{t(t+1)}
\]
Substitute the original expression for \( y \) back into the equation:
\[
\frac{dy}{dt} = \sqrt{\frac{t}{t+1}} \cdot \frac{1}{2} \cdot \frac{1}{t(t+1)}
\]

### Final Answer:
Thus, the derivative of \( y \) with respect to \( t \) is:
\[
\frac{dy}{dt} = \frac{1}{2} \cdot \frac{\sqrt{t}}{(t+1)^{3/2} t}
\]

Let me know if you'd like further clarification or more details on any steps!

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Here are some related questions for further practice:

1. Can you apply logarithmic differentiation to \( y = t^t \)?
2. How does logarithmic differentiation help with products of functions?
3. What is the derivative of \( y = \ln \left( \frac{t^2 + 1}{t+2} \right) \)?
4. How would you differentiate a more complex function like \( y = \frac{(t^3 + 1)^{1/3}}{t^2 - 1} \)?
5. How does the chain rule apply to logarithmic differentiation?

**Tip:** Always look for opportunities to simplify your expressions before differentiating. This can reduce errors and make the differentiation process easier.

In Exercises 45–58, use logarithmic differentiation to find the derivative of y with respect to the given independent variable. y = sqrt(t / (t + 1))

Find the derivative of y = sqrt(t / (t + 1)) using logarithmic differentiation. This solution involves key concepts such as the power rule, chain rule, and properties of logarithms. Suitable for college-level calculus students.

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