Final Answer:
The derivative of the function is y' = (3ln(x) + 2)/(3x+1)
Step-by-step explanation:
Logarithmic differentiation is a technique used to find the derivative of a function that is defined as an exponential or a power function. It works by taking the natural logarithm of both sides of the equation and then differentiating both sides.
Here's a step-by-step explanation of how to use logarithmic differentiation to find the derivative of y=₃√(1)/x(3x+1):
Take the natural logarithm of both sides of the equation:
ln(y) = ln(₃√(1)/x(3x+1))
Use the properties of logarithms to simplify the expression:
ln(y) = -ln(x) - ln(3x+1)
Differentiate both sides of the equation:
d/dx ln(y) = d/dx [-ln(x) - ln(3x+1)]
1/y * dy/dx = -1/x - 3/(3x+1)
Solve for dy/dx:
dy/dx = y * [-1/x - 3/(3x+1)]
Substitute y back into the expression:
dy/dx = (₃√(1)/x(3x+1)) * [-1/x - 3/(3x+1)]
dy/dx = (3ln(x) + 2)/(3x+1)
Therefore, the derivative of the function y=₃√(1)/x(3x+1) using logarithmic differentiation is y' = (3ln(x) + 2)/(3x+1).