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We are tasked with finding the derivative of the function \( y = x^{4x + 3} \) using **logarithmic differentiation**.

Let's go step by step:

### Step 1: Apply logarithmic differentiation

Start by taking the natural logarithm (ln) of both sides of the equation:

\[
\ln y = \ln \left( x^{4x + 3} \right)
\]

Using the logarithmic property \( \ln(a^b) = b \ln(a) \), we simplify the right-hand side:

\[
\ln y = (4x + 3) \ln x
\]

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

Now, differentiate both sides with respect to \(x\). Remember to apply the product rule to the right-hand side:

\[
\frac{d}{dx} \left( \ln y \right) = \frac{d}{dx} \left( (4x + 3) \ln x \right)
\]

Using the chain rule on the left-hand side and the product rule on the right-hand side:

\[
\frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}(4x + 3) \cdot \ln x + (4x + 3) \cdot \frac{d}{dx}(\ln x)
\]

Now, compute the derivatives on the right-hand side:
\[
\frac{d}{dx}(4x + 3) = 4 \quad \text{and} \quad \frac{d}{dx}(\ln x) = \frac{1}{x}
\]

Thus, we have:

\[
\frac{1}{y} \cdot \frac{dy}{dx} = 4 \ln x + \frac{4x + 3}{x}
\]

### Step 3: Solve for \( \frac{dy}{dx} \)

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

\[
\frac{dy}{dx} = y \left( 4 \ln x + \frac{4x + 3}{x} \right)
\]

Finally, substitute \(y = x^{4x + 3}\) back into the equation:

\[
\frac{dy}{dx} = x^{4x + 3} \left( 4 \ln x + \frac{4x + 3}{x} \right)
\]

### Final Result:

\[
\frac{dy}{dx} = x^{4x + 3} \left( 4 \ln x + \frac{4x + 3}{x} \right)
\]

This is the derivative of \(y\) with respect to \(x\) using logarithmic differentiation.

---

Would you like further clarification or details on any of the steps? 

Here are 5 related questions to expand your understanding:
1. How is logarithmic differentiation different from standard differentiation?
2. Why is the product rule necessary when differentiating \( (4x + 3) \ln x \)?
3. Can logarithmic differentiation be applied to functions involving sums of powers?
4. What other types of functions benefit from logarithmic differentiation?
5. How would you differentiate \( y = x^{x^2} \) using logarithmic differentiation?

**Tip**: Logarithmic differentiation is particularly useful when dealing with functions where the variable appears both in the base and the exponent!

Use logarithmic differentiation to find the derivative of y with respect to x. y = x^(4x + 3)

Learn how to find the derivative of y = x^(4x + 3) using logarithmic differentiation. This problem covers key concepts such as logarithmic properties, the product rule, and the chain rule, making it ideal for advanced calculus students.

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