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
- 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].
- Next, apply another logarithm property that allows us to bring the exponents in front of the logarithms: ln[A^n] = n*ln[A].
- 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.
- 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.