https://file.aiquickdraw.com/mathbot/file/17270374666819ozwkgdk.jpg

We are asked to use **logarithmic differentiation** to find the derivative of the function:

\[
y = t(t + 3)(t + 2)
\]

### Steps:

1. **Take the natural logarithm** on both sides of the equation:

\[
\ln(y) = \ln(t) + \ln(t + 3) + \ln(t + 2)
\]

Here, we use the logarithmic property:

\[
\ln(ABC) = \ln(A) + \ln(B) + \ln(C)
\]

2. **Differentiate both sides** with respect to \( t \). Using implicit differentiation on the left-hand side and the chain rule, we get:

\[
\frac{1}{y} \frac{dy}{dt} = \frac{1}{t} + \frac{1}{t + 3} + \frac{1}{t + 2}
\]

3. **Solve for \( \frac{dy}{dt} \)** by multiplying both sides by \( y \):

\[
\frac{dy}{dt} = y \left( \frac{1}{t} + \frac{1}{t + 3} + \frac{1}{t + 2} \right)
\]

4. **Substitute \( y \) back** as \( t(t + 3)(t + 2) \):

\[
\frac{dy}{dt} = t(t + 3)(t + 2) \left( \frac{1}{t} + \frac{1}{t + 3} + \frac{1}{t + 2} \right)
\]

5. **Simplify the expression**:

The terms \( \frac{1}{t} + \frac{1}{t + 3} + \frac{1}{t + 2} \) cannot be further combined without specific values of \( t \), but this is the final derivative in simplified form.

### Final Answer:

\[
\frac{dy}{dt} = t(t + 3)(t + 2) \left( \frac{1}{t} + \frac{1}{t + 3} + \frac{1}{t + 2} \right)
\]

Would you like more details or examples on this topic?

---

Here are 5 related questions to expand on this:

1. What is the advantage of using logarithmic differentiation over the product rule in this case?
2. How would you approach this problem using the product rule instead of logarithmic differentiation?
3. Can logarithmic differentiation be applied to functions involving quotients? How?
4. What is the derivative of \( \ln(x) \), and how is it used in logarithmic differentiation?
5. How would the result change if \( y = t(t+3)(t+2)^2 \)?

**Tip:** Logarithmic differentiation is particularly useful when dealing with products of multiple functions or variables raised to powers.

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable: y = (t)(t + 3)(t + 2).

Learn how to apply logarithmic differentiation to find the derivative of y = t(t + 3)(t + 2). This problem uses the chain rule and natural logarithms to simplify complex products, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation