) Use logarithmic differentiation to find the derivative of
y = 4x sinh(x)
ex2

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

### Step 1: Take the Natural Logarithm of Both Sides

First, let's take the natural logarithm of both sides to simplify differentiation:

\[
\ln(y) = \ln(4x \sinh(x) e^{x^2})
\]

### Step 2: Apply Logarithmic Properties

Use the logarithmic properties to separate the terms:

\[
\ln(y) = \ln(4x) + \ln(\sinh(x)) + \ln(e^{x^2})
\]

Simplify using \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(e^a) = a\):

\[
\ln(y) = \ln(4) + \ln(x) + \ln(\sinh(x)) + x^2
\]

### Step 3: Differentiate Both Sides with Respect to \(x\)

Differentiate both sides of the equation with respect to \(x\). Remember that \(\frac{d}{dx}(\ln(y)) = \frac{y'}{y}\) and apply the chain rule:

\[
\frac{d}{dx}(\ln(y)) = \frac{d}{dx}(\ln(4) + \ln(x) + \ln(\sinh(x)) + x^2)
\]

Simplify the differentiation:

\[
\frac{y'}{y} = 0 + \frac{1}{x} + \frac{1}{\sinh(x)} \cosh(x) + 2x
\]

### Step 4: Solve for \(y'\)

Now, solve for \(y'\) by multiplying both sides by \(y\):

\[
y' = y \left( \frac{1}{x} + \frac{\cosh(x)}{\sinh(x)} + 2x \right)
\]

Since \(y = 4x \sinh(x) e^{x^2}\), substitute \(y\) back into the equation:

\[
y' = 4x \sinh(x) e^{x^2} \left( \frac{1}{x} + \coth(x) + 2x \right)
\]

### Step 5: Simplify the Expression

Now, distribute and simplify the expression:

\[
y' = 4 \sinh(x) e^{x^2} + 4x \coth(x) \sinh(x) e^{x^2} + 8x^2 \sinh(x) e^{x^2}
\]

Since \( \coth(x) \sinh(x) = \cosh(x) \), the expression simplifies to:

\[
y' = 4 \sinh(x) e^{x^2} + 4x \cosh(x) e^{x^2} + 8x^2 \sinh(x) e^{x^2}
\]

This is the derivative \( y' \) of the function \( y = 4x \sinh(x) e^{x^2} \).

### Final Answer

\[
y' = 4 \sinh(x) e^{x^2} + 4x \cosh(x) e^{x^2} + 8x^2 \sinh(x) e^{x^2}
\]

Would you like to see more details or have any questions?

### Related Questions to Explore

1. How do you apply logarithmic differentiation to a product of three functions?
2. What are the properties of hyperbolic functions like \(\sinh(x)\) and \(\cosh(x)\)?
3. How do you differentiate \(y = x^x\) using logarithmic differentiation?
4. What are some common mistakes to avoid when using logarithmic differentiation?
5. How can logarithmic differentiation be useful for functions involving exponentials and logarithms?

**Tip:** When using logarithmic differentiation, remember to apply the chain rule carefully, especially when the function involves products or quotients.

Learn how to find the derivative of y = 4x sinh(x) e^(x^2) using logarithmic differentiation. Detailed step-by-step solution with application of the chain rule and properties of hyperbolic functions. Suitable for advanced undergraduate students studying calculus and differential equations.

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