Math Problem Statement
Use logarithmic differentiation to find the derivative of the function.
y = (e−x cos2(x))/(4x2 + 5x + 6)
Solution
To find the derivative of the given function
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 to both sides:
Using the properties of logarithms:
and , this becomes:
Now, simplify:
Step 2: Differentiate both sides with respect to .
Using implicit differentiation:
Now differentiate the right-hand side:
Step 3: Solve for .
Multiply both sides by to isolate :
Since , substitute it back:
Final Answer:
Do you need further details or have any questions?
Here are some related questions:
- What are the key benefits of logarithmic differentiation?
- How would the solution change if there were a product instead of a quotient?
- Can logarithmic differentiation be applied to implicit functions?
- How do the derivative rules for logarithmic functions differ from exponential ones?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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