Use logarithmic differentiation to find the derivative of the function.

*y* = *(e*−*x* cos2(*x*))/(4*x*2 + 5*x* + 6)

To find the derivative of the given function  
\[
y = \frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6}
\]
using **logarithmic differentiation**, follow these steps:

---

### Step 1: Take the natural logarithm of both sides.
Since logarithmic differentiation simplifies products, quotients, and exponents, we start by applying \(\ln\) to both sides:

\[
\ln(y) = \ln\left(\frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6}\right)
\]

Using the properties of logarithms:  
\(\ln(a / b) = \ln(a) - \ln(b)\) and \(\ln(a^n) = n \ln(a)\), this becomes:

\[
\ln(y) = \ln\left(e^{-x} \cos^2(x)\right) - \ln\left(4x^2 + 5x + 6\right)
\]

\[
\ln(y) = \left(\ln(e^{-x}) + \ln(\cos^2(x))\right) - \ln(4x^2 + 5x + 6)
\]

Now, simplify:

\[
\ln(y) = (-x) + 2 \ln(\cos(x)) - \ln(4x^2 + 5x + 6)
\]

---

### Step 2: Differentiate both sides with respect to \(x\).
Using implicit differentiation:

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

Now differentiate the right-hand side:

\[
\frac{d}{dx}[-x + 2 \ln(\cos(x)) - \ln(4x^2 + 5x + 6)]
\]

\[
\frac{dy}{dx} \cdot \frac{1}{y} = -1 + 2 \cdot \frac{-\sin(x)}{\cos(x)} - \frac{8x + 5}{4x^2 + 5x + 6}
\]

\[
\frac{1}{y} \frac{dy}{dx} = -1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}
\]

---

### Step 3: Solve for \(\frac{dy}{dx}\).
Multiply both sides by \(y\) to isolate \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = y \left(-1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}\right)
\]

Since \(y = \frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6}\), substitute it back:

\[
\frac{dy}{dx} = \frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6} \left(-1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}\right)
\]

---

### Final Answer:
\[
\boxed{\frac{dy}{dx} = \frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6} \left(-1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}\right)}
\]

---

Do you need further details or have any questions? 

Here are some related questions:
1. What are the key benefits of logarithmic differentiation?
2. How would the solution change if there were a product instead of a quotient?
3. Can logarithmic differentiation be applied to implicit functions?
4. How do the derivative rules for logarithmic functions differ from exponential ones?
5. What are other scenarios where taking logarithms simplifies differentiation?

**Tip:** Logarithmic differentiation is especially helpful when dealing with complex products, quotients, or variables raised to variables.

Use logarithmic differentiation to find the derivative of the function y = (e^(-x) * cos^2(x)) / (4x^2 + 5x + 6).

Learn how to apply logarithmic differentiation to find the derivative of the function y = (e^(-x) * cos^2(x)) / (4x^2 + 5x + 6). This solution covers key concepts like implicit differentiation, properties of logarithms, and trigonometric derivatives. Perfect for high school or early college students studying advanced calculus.

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