Math Problem Statement

Use logarithmic differentiation to find the derivative of the function.

y = (ex cos2(x))/(4x2 + 5x + 6)

Solution

To find the derivative of the given function
y=excos2(x)4x2+5x+6y = \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\ln to both sides:

ln(y)=ln(excos2(x)4x2+5x+6)\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)\ln(a / b) = \ln(a) - \ln(b) and ln(an)=nln(a)\ln(a^n) = n \ln(a), this becomes:

ln(y)=ln(excos2(x))ln(4x2+5x+6)\ln(y) = \ln\left(e^{-x} \cos^2(x)\right) - \ln\left(4x^2 + 5x + 6\right)

ln(y)=(ln(ex)+ln(cos2(x)))ln(4x2+5x+6)\ln(y) = \left(\ln(e^{-x}) + \ln(\cos^2(x))\right) - \ln(4x^2 + 5x + 6)

Now, simplify:

ln(y)=(x)+2ln(cos(x))ln(4x2+5x+6)\ln(y) = (-x) + 2 \ln(\cos(x)) - \ln(4x^2 + 5x + 6)


Step 2: Differentiate both sides with respect to xx.

Using implicit differentiation:

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

Now differentiate the right-hand side:

ddx[x+2ln(cos(x))ln(4x2+5x+6)]\frac{d}{dx}[-x + 2 \ln(\cos(x)) - \ln(4x^2 + 5x + 6)]

dydx1y=1+2sin(x)cos(x)8x+54x2+5x+6\frac{dy}{dx} \cdot \frac{1}{y} = -1 + 2 \cdot \frac{-\sin(x)}{\cos(x)} - \frac{8x + 5}{4x^2 + 5x + 6}

1ydydx=12tan(x)8x+54x2+5x+6\frac{1}{y} \frac{dy}{dx} = -1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}


Step 3: Solve for dydx\frac{dy}{dx}.

Multiply both sides by yy to isolate dydx\frac{dy}{dx}:

dydx=y(12tan(x)8x+54x2+5x+6)\frac{dy}{dx} = y \left(-1 - 2 \tan(x) - \frac{8x + 5}{4x^2 + 5x + 6}\right)

Since y=excos2(x)4x2+5x+6y = \frac{e^{-x} \cos^2(x)}{4x^2 + 5x + 6}, substitute it back:

dydx=excos2(x)4x2+5x+6(12tan(x)8x+54x2+5x+6)\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:

dydx=excos2(x)4x2+5x+6(12tan(x)8x+54x2+5x+6)\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.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Differentiation
Implicit Differentiation
Exponential Functions
Trigonometric Functions
Quotient Rule

Formulas

ln(a / b) = ln(a) - ln(b)
ln(a^n) = n * ln(a)
d/dx[ln(y)] = (1/y) * dy/dx
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec^2(x)

Theorems

Properties of Logarithms
Derivative Rules for Trigonometric Functions
Derivative Rules for Exponential Functions

Suitable Grade Level

Grades 11-12 / Early College