Final answer:
To find dy/dx using logarithmic differentiation, take the natural logarithm of both sides of the equation and apply logarithmic properties. Then, differentiate both sides using the chain rule.
Step-by-step explanation:
To find dy/dx using logarithmic differentiation, we first take the natural logarithm of both sides of the equation y = (x + 1)(x - 10)/(x - 1)(x + 10):
ln(y) = ln((x + 1)(x - 10)/(x - 1)(x + 10))
Next, we use logarithmic properties to simplify the expression:
ln(y) = ln(x + 1) + ln(x - 10) - ln(x - 1) - ln(x + 10)
To differentiate both sides with respect to x, we use the chain rule:
(1/y) * (dy/dx) = (1/(x + 1)) + (1/(x - 10)) - (1/(x - 1)) - (1/(x + 10))
Multiplying both sides by y gives us:
dy/dx = y * [(1/(x + 1)) + (1/(x - 10)) - (1/(x - 1)) - (1/(x + 10))]