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

To find the derivative of the function \( f(x) = \frac{4x \sinh(x)}{e^{x^2}} \) using logarithmic differentiation, follow these steps:

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

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

### Step 3: Differentiate both sides with respect to \( x \)
Differentiate implicitly with respect to \( x \):
\[
\frac{1}{f(x)} \cdot f'(x) = \frac{d}{dx}[\ln(4x)] + \frac{d}{dx}[\ln(\sinh(x))] - \frac{d}{dx}[x^2]
\]

### Step 4: Compute the derivatives on the right-hand side
Differentiate each term individually:

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

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

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

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

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

### 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 simplifies to:
\[
f'(x) = \frac{4 \sinh(x) (1 + x \coth(x) - 2x^2)}{e^{x^2}}
\]

Would you like further details or have any questions?

### Related Questions
1. How does the hyperbolic function \(\sinh(x)\) behave compared to the trigonometric sine function?
2. What is the significance of using logarithmic differentiation in complex functions?
3. Can you explain the behavior of the function \( \coth(x) \) as \( x \) approaches 0?
4. How would the derivative change if the exponent in the denominator were different, e.g., \( e^{2x^2} \)?
5. What is the practical application of differentiating hyperbolic functions?

### Tip
When using logarithmic differentiation, always simplify the expression inside the logarithm first to make the differentiation process easier.

Explore how to find the derivative of the function (4xsinh(x))/e^x^2 using logarithmic differentiation. Learn step-by-step with detailed explanations and formulas. Suitable for advanced college-level students and beyond.

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