Question

Perform logarithmic differentiation on the expression y = cuberoot(x²).

a) 2/(3x)
b) 1/(3x)
c) 2/(3x²)
d) 1/(2*cuberoot(x))

Answer 1

Logarithmic differentiation of y = cuberoot(x²) involves applying the properties of logarithms to differentiate y with respect to x, resulting in the derivative dy/dx = 2/(3x), which corresponds to option a) 2/(3x). The correct answer is a) 2/(3x).

Step-by-step explanation:

To perform logarithmic differentiation on the expression y = cuberoot(x²), we apply the following steps:

1. Express the equation y = x²^(1/3).
2. Take the natural logarithm of both sides to get ln(y) = ln(x²^(1/3)).
3. Apply the power rule of logarithms to simplify the right-hand side: ln(y) = (1/3)ln(x²).
4. Further simplify the right-hand side using the chain rule for logarithms: ln(y) = (2/3)ln(x).
5. Now, differentiate both sides with respect to x. Remembering that d/dx[ln(x)] = 1/x and using the chain rule on the left side gives us (1/y)(dy/dx) = (2/3)(1/x).
6. Solve for dy/dx to find the derivative of y with respect to x: dy/dx = (2/3x)y.
7. Substitute back the original y to get the final answer dy/dx = (2/3x)(x²^(1/3)).
8. Finally, simplify to get the answer: dy/dx = 2/(3x).

Therefore, the correct answer is a) 2/(3x).