https://file.aiquickdraw.com/mathbot/file/1724408173602j5rsm12s.jpg

مشتق عبارت داخل عکس با راه حل تشریحی بگیر

The expression given in the image is:

\[
Y = \left[\ln(3x - 1)\right]^2
\]

To find the derivative of \( Y \) with respect to \( x \), we will use the chain rule. Let's go through the steps:

### Step 1: Define the inner function and the outer function.
Let:
\[
u = \ln(3x - 1)
\]
So the function becomes:
\[
Y = u^2
\]

### Step 2: Differentiate the outer function with respect to \( u \).
The derivative of \( Y = u^2 \) with respect to \( u \) is:
\[
\frac{dY}{du} = 2u
\]

### Step 3: Differentiate the inner function \( u = \ln(3x - 1) \) with respect to \( x \).
The derivative of \( u = \ln(3x - 1) \) with respect to \( x \) is:
\[
\frac{du}{dx} = \frac{1}{3x - 1} \times 3 = \frac{3}{3x - 1}
\]

### Step 4: Apply the chain rule.
Using the chain rule, the derivative of \( Y \) with respect to \( x \) is:
\[
\frac{dY}{dx} = \frac{dY}{du} \times \frac{du}{dx} = 2u \times \frac{3}{3x - 1}
\]

### Step 5: Substitute back the value of \( u \).
Since \( u = \ln(3x - 1) \), we get:
\[
\frac{dY}{dx} = 2 \ln(3x - 1) \times \frac{3}{3x - 1}
\]

### Final Answer:
\[
\frac{dY}{dx} = \frac{6 \ln(3x - 1)}{3x - 1}
\]

This is the derivative of the given expression \( Y = \left[\ln(3x - 1)\right]^2 \) with respect to \( x \).

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Do you want details or have any questions?

Here are some related questions you might find useful:
1. What is the derivative of \( Y = [\ln(2x + 4)]^2 \)?
2. How would you find the second derivative of the given function?
3. What is the domain of the function \( Y = \left[\ln(3x - 1)\right]^2 \)?
4. How would the derivative change if the original function was \( Y = \left[\ln(3x^2 - 1)\right]^2 \)?
5. How do logarithmic and exponential functions relate to each other in differentiation?

**Tip:** When differentiating functions involving logarithms, always remember to apply the chain rule, especially when the argument of the logarithm is more complex than a simple variable.

Understand how to find the derivative of Y = [ln(3x - 1)]^2 with detailed steps. Learn the application of the chain rule in differentiation of logarithmic functions. Suitable for advanced high school students and calculus enthusiasts.

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