Final answer:
To find the derivative of f(x)=xex²(x^2+1)^10 using logarithmic differentiation, you can follow these steps: Step 1: Take the natural logarithm of both sides of the equation. Step 2: Use the properties of logarithms to simplify the expression. Step 3: Differentiate both sides of the equation with respect to x.
Step-by-step explanation:
Derivative of f(x)=xex²(x^2+1)^10 using logarithmic differentiation:
Step 1: Take the natural logarithm of both sides of the equation to simplify the expression:
ln(f(x)) = ln(xex²(x^2+1)^10)
Step 2: Use the properties of logarithms to simplify further:
ln(f(x)) = ln(x) + ln(ex²) + ln((x^2+1)^10)
Step 3: Differentiate both sides of the equation with respect to x:
(1/f(x)) * f'(x) = 1/x + 2x + 10(x^2+1) * (2x/(x^2+1))
Step 4: Solve for f'(x) by multiplying both sides by f(x):
f'(x) = f(x) * (1/x + 2x + 20x(x^2+1)/(x^2+1))
Step 5: Substitute f(x) back in:
f'(x) = xex²(x^2+1)^10 * (1/x + 2x + 20x(x^2+1)/(x^2+1))
Step 6: Simplify the expression:
f'(x) = xex²(x^2+1)^10 * (1/x + 2x + 20x)
Step 7: Further simplify and combine like terms:
f'(x) = xex²(x^2+1)^10 * (1/x + 22x)
Therefore, the derivative of f(x)=xex²(x^2+1)^10 using logarithmic differentiation is f'(x) = xex²(x^2+1)^10 * (1/x + 22x).