Use logarithmic differentiation to find the following derivative:

y=(x+6)^4/(x-5)^6. Please show all step so I can understand for future reference.

ln y = ln(x+6)^4 - ln(x-5)^6

ln y = 4ln(x+6) - 6ln(x-5)

y' / y = 4/(x+6) - 6/(x-5)

y' or dy/dx = y(4/(x+6) - 6/(x-5))
or
(x+6)^4/(x-5)^6 (4/(x+6) - 6/(x-5) )

To find the derivative of the function y = (x + 6)^4 / (x - 5)^6 using logarithmic differentiation, follow these steps:

Step 1: Take the natural logarithm (ln) of both sides of the equation to convert the function into a logarithmic form:
ln(y) = ln((x + 6)^4 / (x - 5)^6)

Step 2: Apply the logarithmic rules to simplify the expression:
ln(y) = 4ln(x + 6) - 6ln(x - 5)

Step 3: Differentiate both sides of the equation with respect to x using the chain rule:
(d/dx) ln(y) = (d/dx) (4ln(x + 6) - 6ln(x - 5))

To differentiate ln(y), recall that d(ln(y))/dx = 1/y * dy/dx. So:
1/y * dy/dx = 4 * 1/(x + 6) - 6 * 1/(x - 5)

Step 4: Multiply both sides by y to make dy/dx the subject of the equation:
dy/dx = y * (4/(x + 6) - 6/(x - 5))

Since y = (x + 6)^4 / (x - 5)^6, substitute this value back into the equation:
dy/dx = (x + 6)^4 / (x - 5)^6 * (4/(x + 6) - 6/(x - 5))

This is the derivative of the original function y = (x + 6)^4 / (x - 5)^6 using logarithmic differentiation.