Use logarithmic differentiation to find the derivative of (4xsinh(x))/e^x^2

Let's use logarithmic differentiation to find the derivative of the function \( f(x) = \frac{4x \sinh(x)}{e^{x^2}} \).

### Step 1: Take the natural logarithm of both sides
Take the natural logarithm of both sides:
\[
\ln(f(x)) = \ln\left(\frac{4x \sinh(x)}{e^{x^2}}\right)
\]

### Step 2: Apply logarithm properties
Use logarithm properties to simplify the expression:
\[
\ln(f(x)) = \ln(4x \sinh(x)) - \ln(e^{x^2})
\]
\[
\ln(f(x)) = \ln(4x) + \ln(\sinh(x)) - x^2
\]

### Step 3: Differentiate both sides with respect to \( x \)
Differentiate both sides with respect to \( x \), using the chain rule on the left side:
\[
\frac{1}{f(x)} \cdot f'(x) = \frac{d}{dx}[\ln(4x)] + \frac{d}{dx}[\ln(\sinh(x))] - \frac{d}{dx}[x^2]
\]

### Step 4: Differentiate each term on the right-hand side
Now, calculate the derivatives of the terms on the right:

1. **For \( \ln(4x) \)**:
   \[
   \frac{d}{dx}[\ln(4x)] = \frac{1}{4x} \cdot 4 = \frac{1}{x}
   \]

2. **For \( \ln(\sinh(x)) \)**:
   \[
   \frac{d}{dx}[\ln(\sinh(x))] = \frac{1}{\sinh(x)} \cdot \cosh(x) = \coth(x)
   \]

3. **For \( x^2 \)**:
   \[
   \frac{d}{dx}[x^2] = 2x
   \]

### Step 5: Combine the results
Substitute these into the equation:
\[
\frac{1}{f(x)} \cdot f'(x) = \frac{1}{x} + \coth(x) - 2x
\]

### Step 6: Solve for \( f'(x) \)
Multiply both sides by \( f(x) = \frac{4x \sinh(x)}{e^{x^2}} \) to isolate \( f'(x) \):
\[
f'(x) = \frac{4x \sinh(x)}{e^{x^2}} \left(\frac{1}{x} + \coth(x) - 2x\right)
\]

### Final Answer
The derivative of the function \( f(x) = \frac{4x \sinh(x)}{e^{x^2}} \) is:
\[
f'(x) = \frac{4 \sinh(x)}{e^{x^2}} + \frac{4x \cosh(x)}{e^{x^2}} - \frac{8x^2 \sinh(x)}{e^{x^2}}
\]

This can be written as:
\[
f'(x) = \frac{4 \sinh(x) (1 + x \coth(x) - 2x^2)}{e^{x^2}}
\]

Would you like further details or have any questions?

### Related Questions
1. What is the significance of the hyperbolic function \( \sinh(x) \) in calculus?
2. How does logarithmic differentiation simplify finding derivatives of products and quotients?
3. What are the properties of the hyperbolic cotangent function \( \coth(x) \)?
4. How would the derivative change if the function was \( \frac{4x \cosh(x)}{e^{x^2}} \) instead?
5. Can logarithmic differentiation be applied to functions involving trigonometric functions?

### Tip
When dealing with products or quotients of functions, logarithmic differentiation can significantly simplify the differentiation process, especially when the functions are complicated.

Learn how to find the derivative of the function (4x sinh(x)) / e^(x^2) using logarithmic differentiation. This detailed solution covers the application of logarithmic differentiation, chain rule, and derivatives of hyperbolic functions. Suitable for advanced undergraduate students and learners interested in calculus and mathematical analysis.

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