33.6k views
0 votes
Let f(x) = x^6 (x-3)^8/(x^2+8)^2 Use logarithmic differentiation to determine the derivative.

1 Answer

6 votes

Answer: f'(x) = x^6 (x-3)^8/(x^2+8)^2 * (6/x + 8/(x-3) - 4x/(x^2+8))
Step-by-step explanation:


Given: f(x) = x^6(x-3)^8/(x^2+8)^2

To find the derivative of f(x) = x^6(x-3)^8/(x^2+8)^2 using logarithmic differentiation, follow these steps:

1. Take the natural logarithm (ln) of both sides of the equation:
ln(f(x)) = ln(x^6(x-3)^8/(x^2+8)^2)

2. Apply the logarithmic properties to simplify the expression:
ln(f(x)) = 6ln(x) + 8ln(x-3) - 2ln(x^2+8)

3. Differentiate both sides of the equation with respect to x using the chain rule:
f'(x)/f(x) = 6/x + 8/(x-3) - 4x/(x^2+8)

4. Finally, multiply both sides by f(x) to find f'(x):
f'(x) = f(x) * (6/x + 8/(x-3) - 4x/(x^2+8))
f'(x) = x^6 (x-3)^8/(x^2+8)^2 * (6/x + 8/(x-3) - 4x/(x^2+8))

That's the derivative of the given function using logarithmic differentiation.

User Elec
by
7.6k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.