find the derivative of y =(x+3)^3 / sqrt(x^2 -1) using logarithmic differentation.

y =(x+3)^3 / √(x^2 -1)

lny = 3ln(x+3) - 1/2 ln(x^2-1)
1/y y' = 3/(x+3) - x/(x^2-1)
1/y y' = (3(x^2-1) - x(x+3))/((x+3)(x^2-1))
1/y y' = (2x^2-3x-3)/((x+3)(x^2-1))

y' = (x+3)^3 / √(x^2 -1) * ((2x^2-3x-3)/((x+3)(x^2-1)))
= ((x+3)^2 * (2x^2-3x-3))/(x^2-1)^(3/2)

To find the derivative of the given function using logarithmic differentiation, follow these steps:

Step 1: Write the given function in the logarithmic form.
Take the natural logarithm of both sides of the equation:
ln(y) = ln((x+3)^3 / sqrt(x^2 - 1))

Step 2: Simplify using logarithmic properties.
Apply the power rule for logarithms:
ln(y) = 3ln(x+3) - 1/2ln(x^2 - 1)

Step 3: Differentiate implicitly.
Differentiate both sides of the equation with respect to x:
1/y * dy/dx = d/dx(3ln(x+3)) - d/dx(1/2ln(x^2 - 1))

Step 4: Simplify and solve for dy/dx.
Simplify the right side of the equation using the chain rule and the derivative of natural logarithm. The derivative of ln(x) with respect to x is 1/x. Applying these rules, we get:
1/y * dy/dx = 3 * 1/(x + 3) - 1/2 * 2x / (x^2 - 1)

Next, simplify further:
1/y * dy/dx = 3/(x + 3) - x/(x^2 - 1)

Finally, multiply both sides of the equation by y to isolate dy/dx:
dy/dx = y * [3/(x + 3) - x/(x^2 - 1)]

However, we need to substitute the value of y back into the equation since we are given y in terms of x.

y = (x+3)^3 / sqrt(x^2 -1)

Substituting this value into the equation:
dy/dx = [(x+3)^3 / sqrt(x^2 -1)] * [3/(x + 3) - x/(x^2 - 1)]

Simplifying this expression gives the derivative of y with respect to x using logarithmic differentiation.