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The question asks to find the derivative of the function \( f \), defined as:

\[
f(x) = -9x^2 \ln(x)
\]

### Solution:

To find the derivative of the function \( f(x) = -9x^2 \ln(x) \), we need to use the **product rule** because the function is the product of two functions:

1. \( u(x) = -9x^2 \)
2. \( v(x) = \ln(x) \)

The product rule for differentiation states:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]

#### Step 1: Differentiate \( u(x) = -9x^2 \)

\[
u'(x) = -18x
\]

#### Step 2: Differentiate \( v(x) = \ln(x) \)

\[
v'(x) = \frac{1}{x}
\]

#### Step 3: Apply the product rule

\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
\[
f'(x) = (-18x)(\ln(x)) + (-9x^2)\left(\frac{1}{x}\right)
\]

Simplify the second term:

\[
f'(x) = -18x \ln(x) - 9x
\]

So, the derivative of \( f(x) \) is:

\[
f'(x) = -18x \ln(x) - 9x
\]

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Would you like more details on this, or do you have any questions? Here are five related questions for further practice:

1. How do you apply the product rule when there are more than two functions involved?
2. What is the derivative of \( x^3 \ln(x) \)?
3. Can you explain how the chain rule could be used in combination with the product rule?
4. What is the derivative of \( x^2 \ln(x^2) \)?
5. How would you find the second derivative of the given function \( f(x) = -9x^2 \ln(x) \)?

### Tip: 
Always check for simplification opportunities after applying the product rule, especially when combining like terms.

Quelle est la dérivée de la fonction f(x) = -9x^2 ln(x) ?

Learn how to differentiate the function f(x) = -9x^2 ln(x) by applying the product rule and logarithmic differentiation. This step-by-step solution is designed for students studying advanced calculus in Grades 11-12 or at the university level.

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