Question

Find \(d y / d x\) using logarithmic differentiation. $$ y=\frac{(x+1)(x+2)}{(x-1)(x-2)} $$

Short answer

The derivative of the function, \( dy/dx \), is \( ((x+1)(x+2))/((x-1)(x-2)) * ((1/(x+1)) + (1/(x+2)) - (1/(x-1)) - (1/(x-2))) \)

Step-by-step solution

Express the function using natural logarithm

Express the given function in terms of the natural logarithm \( ln \). This will simplify the function and make it easier to find its derivative. Since the natural logarithm of \( y = ln(y) = ln((x+1)(x+2)) - ln((x-1)(x-2)) \)

Use the logarithmic properties

The natural logarithm has a property that it can separate the terms that are being multiplied or divided. Apply this property: \( ln(y) = ln(x+1) + ln(x+2) - ln(x-1) - ln(x-2) \)

Differentiate both sides

It's easier to find the derivative of the function now. Differentiate both sides with respect to \(x\). This gives: \( 1/y * dy/dx = (1/(x+1)) + (1/(x+2)) - (1/(x-1)) - (1/(x-2)) \)

Solve for dy/dx

Finally, isolate \( dy/dx \) by multiplying both sides by \( y \). Given that \( y = ((x+1)(x+2))/((x-1)(x-2)) \), substituting this in gives the derivative \( dy/dx = ((x+1)(x+2))/((x-1)(x-2)) * ((1/(x+1)) + (1/(x+2)) - (1/(x-1)) - (1/(x-2))) \)