Use logarithmic differentiation to find the following derivative
y=(x+6)^4/(x-5)^6.
Please show all steps so I can follow for future understanding.
Thanks.
4 log (x+6) - 6 log (x-5)
I had done this question or you yesterday
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Always check back to your previous post to avoid duplication of solutions.
To find the derivative of the given function, we can use logarithmic differentiation. Here are the step-by-step instructions on how to do it:
Step 1: Take the natural logarithm of both sides of the equation.
ln(y) = ln((x+6)^4/(x-5)^6)
Step 2: Apply logarithmic properties to simplify the equation.
ln(y) = 4ln(x+6) - 6ln(x-5)
Step 3: Differentiate both sides of the equation with respect to x.
d/dx[ln(y)] = d/dx[4ln(x+6) - 6ln(x-5)]
Step 4: Use the chain rule and the derivative of ln(x).
1/y * dy/dx = 4/(x+6) - 6/(x-5)
Step 5: Multiply both sides by y to get rid of the denominator.
dy/dx = y * (4/(x+6) - 6/(x-5))
Step 6: Substitute the original function back into the equation.
dy/dx = (x+6)^4/(x-5)^6 * (4/(x+6) - 6/(x-5))
Therefore, the derivative of y=(x+6)^4/(x-5)^6 with respect to x is dy/dx = (x+6)^4/(x-5)^6 * (4/(x+6) - 6/(x-5)).