For the function

*y=ln\sqrt[5]{x^3}*, find *dydx*.

We are tasked with finding the derivative \( \frac{dy}{dx} \) of the function:

\[
y = \ln \sqrt[5]{x^3}
\]

### Step 1: Simplify the expression inside the logarithm

Recall that:

\[
\sqrt[5]{x^3} = x^{\frac{3}{5}}
\]

Thus, the function becomes:

\[
y = \ln \left( x^{\frac{3}{5}} \right)
\]

### Step 2: Apply the logarithmic property

We can use the property of logarithms \( \ln(a^b) = b \ln(a) \). Therefore:

\[
y = \frac{3}{5} \ln(x)
\]

### Step 3: Differentiate with respect to \( x \)

Now, we differentiate \( y = \frac{3}{5} \ln(x) \) using the derivative of \( \ln(x) \), which is \( \frac{1}{x} \):

\[
\frac{dy}{dx} = \frac{3}{5} \cdot \frac{1}{x}
\]

### Final Answer:

\[
\frac{dy}{dx} = \frac{3}{5x}
\]

---

Would you like further details or have any questions about this? Here are some related questions for exploration:

1. How do you differentiate other logarithmic functions like \( \ln(x^n) \)?
2. Can you apply the chain rule when the function inside the logarithm is more complex?
3. How do you handle derivatives of logarithmic functions involving constants and exponents?
4. What happens if the function inside the logarithm is a product or quotient of terms?
5. How can you simplify logarithmic expressions to make differentiation easier?

**Tip**: When differentiating logarithmic functions, always look for opportunities to simplify the argument using logarithmic properties first.

For the function y=ln(√[5]{x^3}), find dydx.

Learn how to find the derivative of y = ln(√[5]{x^3}) with a detailed step-by-step solution. This problem involves logarithmic differentiation, applying logarithmic properties, and using basic derivative rules. Suitable for students in Grades 11-12.

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