https://file.aiquickdraw.com/mathbot/file/1727818358705q6wrs4jc.jpg

We are tasked with using **logarithmic differentiation** to find the derivative of the function:

\[
g(x) = \left( \sin(2x) \right)^x
\]

### Step-by-Step Solution

1. **Take the natural logarithm (ln) of both sides**:

   \[
   \ln(g(x)) = \ln\left( (\sin(2x))^x \right)
   \]

   Using the logarithmic property \(\ln(a^b) = b\ln(a)\), this simplifies to:

   \[
   \ln(g(x)) = x \ln(\sin(2x))
   \]

2. **Differentiate both sides with respect to \(x\)** using implicit differentiation. Keep in mind that the derivative of \(\ln(g(x))\) is \(\frac{g'(x)}{g(x)}\).

   \[
   \frac{d}{dx} \left( \ln(g(x)) \right) = \frac{d}{dx} \left( x \ln(\sin(2x)) \right)
   \]

   The left side becomes:

   \[
   \frac{g'(x)}{g(x)}
   \]

   For the right side, use the **product rule**: \( \frac{d}{dx} \left( x \ln(\sin(2x)) \right) = \ln(\sin(2x)) \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(\ln(\sin(2x))) \).

   This simplifies to:

   \[
   \ln(\sin(2x)) + x \cdot \frac{1}{\sin(2x)} \cdot \frac{d}{dx}(\sin(2x))
   \]

3. **Differentiate \(\sin(2x)\)**:

   \[
   \frac{d}{dx}(\sin(2x)) = 2\cos(2x)
   \]

   So the expression becomes:

   \[
   \ln(\sin(2x)) + x \cdot \frac{2\cos(2x)}{\sin(2x)}
   \]

4. **Substitute this result back into the equation**:

   \[
   \frac{g'(x)}{g(x)} = \ln(\sin(2x)) + 2x \cdot \cot(2x)
   \]

5. **Solve for \(g'(x)\)**:

   Multiply both sides by \(g(x) = (\sin(2x))^x\):

   \[
   g'(x) = (\sin(2x))^x \left( \ln(\sin(2x)) + 2x \cdot \cot(2x) \right)
   \]

Thus, the derivative of \( g(x) = (\sin(2x))^x \) is:

\[
g'(x) = (\sin(2x))^x \left( \ln(\sin(2x)) + 2x \cdot \cot(2x) \right)
\]

Would you like further details or have any other questions?

---

Here are 5 related questions to expand on this topic:
1. How is logarithmic differentiation useful when differentiating functions of the form \( f(x)^g(x) \)?
2. What are some common logarithmic properties that simplify complex derivatives?
3. How would the derivative change if the base of the power was another trigonometric function, such as \( \cos(2x) \)?
4. Can logarithmic differentiation be applied to implicit functions? If yes, how?
5. What are the differences between logarithmic differentiation and regular differentiation in terms of steps?

**Tip**: Logarithmic differentiation is particularly effective when both the base and the exponent are functions of \(x\), as it simplifies the application of the product and chain rules.

Use logarithmic differentiation to find the derivative of g(x) = (sin(2x))^x.

Discover how to use logarithmic differentiation to find the derivative of g(x) = (sin(2x))^x. This solution involves key concepts such as logarithmic properties, the product rule, and implicit differentiation. Ideal for calculus students looking for detailed solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation