Use logarithmic differentiation to find 
dy
dx
.
y = x

x2 + 4
,    x > 0
dy
dx
 = 
  
Incorrect: Your answer is incorrect.

To find \(\frac{dy}{dx}\) using logarithmic differentiation for the function:

\[
y = x^{x^2 + 4}
\]

we proceed as follows:

### Step 1: Take the natural logarithm of both sides.

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

Using the logarithmic property \(\ln(a^b) = b \ln a\), we get:

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

### Step 2: Differentiate both sides implicitly with respect to \(x\).

On the left-hand side, by implicit differentiation:

\[
\frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx}
\]

On the right-hand side, use the product rule to differentiate \((x^2 + 4) \ln x\). Let \(u = x^2 + 4\) and \(v = \ln x\), then:

\[
\frac{d}{dx}\left((x^2 + 4)\ln x\right) = \frac{d}{dx}(u v) = \frac{du}{dx}v + u\frac{dv}{dx}
\]

Now, compute each derivative:

- \(\frac{du}{dx} = 2x\)
- \(\frac{dv}{dx} = \frac{1}{x}\)

So,

\[
\frac{d}{dx}\left((x^2 + 4)\ln x\right) = 2x \ln x + (x^2 + 4) \cdot \frac{1}{x}
\]

Simplify the second term:

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

### Step 3: Substitute into the implicit differentiation equation.

Now, we have:

\[
\frac{1}{y} \frac{dy}{dx} = 2x \ln x + x + \frac{4}{x}
\]

### Step 4: Solve for \(\frac{dy}{dx}\).

Multiply both sides by \(y\) to isolate \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = y\left(2x \ln x + x + \frac{4}{x}\right)
\]

### Step 5: Substitute \(y = x^{x^2 + 4}\) back into the equation.

\[
\frac{dy}{dx} = x^{x^2 + 4}\left(2x \ln x + x + \frac{4}{x}\right)
\]

This is the derivative of the given function.

---

Let me know if you'd like further clarification or more details.

### Related Questions:
1. How can logarithmic differentiation simplify complex exponentials?
2. What is the product rule, and how does it apply in this case?
3. Can logarithmic differentiation be applied to non-exponential functions?
4. What are the benefits of implicit differentiation in finding derivatives?
5. How does the chain rule play a role in logarithmic differentiation?

### Tip:
For complex exponentials like \(x^{x^2 + 4}\), logarithmic differentiation often simplifies the process, avoiding messy power rule applications.

Use logarithmic differentiation to find dy/dx. y = x^(x^2 + 4), x > 0

Learn how to apply logarithmic differentiation to find the derivative of y = x^(x^2 + 4). This solution covers key calculus concepts like the product rule, chain rule, and implicit differentiation. Suitable for high school students in grades 11-12.

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x2 + 4 , x > 0 dy dx

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