Final answer:
To find dy/dx using logarithmic differentiation, take the natural logarithm of both sides of the equation, differentiate implicitly, and substitute the expression for y.
Step-by-step explanation:
To find dy/dx using logarithmic differentiation, we can take the natural logarithm (ln) of both sides of the equation y = 4x^(2/3) / (x + 1).
ln(y) = ln(4x^(2/3) / (x + 1))
Using the logarithmic property, ln(a/b) = ln(a) - ln(b), we can rewrite the equation as:
ln(y) = ln(4x^(2/3)) - ln(x + 1)
Now, we can differentiate both sides of the equation implicitly with respect to x:
(1/y) * dy/dx = (2/3) * (1/x) - (1/(x + 1))
Multiplying both sides by y gives:
dy/dx = y * [(2/3) * (1/x) - (1/(x + 1))]
Finally, substituting y = 4x^(2/3) / (x + 1):
dy/dx = 4x^(2/3) / (x + 1) * [(2/3) * (1/x) - (1/(x + 1))]