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Find the derivative of ln[x³(x−9)³ (x²−3)⁶]

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Final answer:

To find the derivative of ln[x³(x-9)³ (x²-3)⁶], one must use logarithmic differentiation by separating the ln of a product into a sum of lns and bringing down the exponents, and then take the derivative term by term.

Step-by-step explanation:

The student has asked to find the derivative of the function ln[x³(x−9)³ (x²−3)⁶]. To do this, we can use the rules of logarithms and derivatives to simplify the problem.

Steps to Find the Derivative

  1. First, apply the property of logarithms that allows us to separate the logarithm of a product into the sum of logarithms: ln[AB] = ln[A] + ln[B]. For our function, we have ln[x³(x−9)³ (x²−3)⁶] = ln[x³] + ln[(x−9)³] + ln[(x²−3)⁶].
  2. Next, apply another logarithm property that allows us to bring the exponents in front of the logarithms: ln[A^n] = n*ln[A].
  3. Now, we can take the derivative of each term separately using the chain rule. Remember, the derivative of ln[x] with respect to x is 1/x, and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
  4. After taking the derivative of each term, we add them together to get the final derivative.

Therefore, by following these steps, the derivative of the function ln[x³(x−9)³ (x²−3)⁶] is obtained as:

3/x + 3/(x-9) + 12/(x²-3)

This assumes that x is within the domain where the expression inside the natural logarithm is positive.

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