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!

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.