Use logarithmic differentiation to determine the derivative of the function defined by
f(x)=x^5(x-3)^9/(x^2+2)^4
To find the derivative of the given function using logarithmic differentiation, follow the steps below:
Step 1: Take the natural logarithm (ln) of both sides of the equation.
ln(f(x)) = ln(x^5(x-3)^9/(x^2+2)^4)
Step 2: Apply the logarithmic property to simplify the expression.
ln(f(x)) = ln(x^5) + ln((x-3)^9) - ln((x^2+2)^4)
ln(f(x)) = 5ln(x) + 9ln(x-3) - 4ln(x^2+2)
Step 3: Differentiate both sides of the equation with respect to x.
d/dx[ln(f(x))] = d/dx[5ln(x) + 9ln(x-3) - 4ln(x^2+2)]
Note: The derivative of ln(f(x)) can be written as f'(x)/f(x) according to the chain rule.
f'(x)/f(x) = d/dx[5ln(x)] + d/dx[9ln(x-3)] - d/dx[4ln(x^2+2)]
Step 4: Simplify the derivatives on the right side of the equation.
f'(x)/f(x) = 5(1/x) + 9(1/(x-3)) - 4(1/(x^2+2))
Step 5: Multiply both sides of the equation by f(x) to isolate f'(x).
f'(x) = f(x)[5(1/x) + 9(1/(x-3)) - 4(1/(x^2+2))]
Now, we substitute the original function f(x) = x^5(x-3)^9/(x^2+2)^4 into the equation to find the derivative.
f'(x) = [x^5(x-3)^9/(x^2+2)^4][5(1/x) + 9(1/(x-3)) - 4(1/(x^2+2))]