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The problem you've provided concerns a function \( f \) defined by:

\[
f(x) = \frac{1}{(x+1)^3}
\]

Here is a breakdown of the tasks:

1. **Recognize that \( f \) is of the form \( u^{-1} \), with \( u = (x + 1)^3 \):**

   - This step asks to express the function \( f(x) \) as \( u^{-1} \) where \( u = (x+1)^3 \). Thus, we define \( u(x) = (x+1)^3 \), and therefore:

     \[
     f(x) = \frac{1}{u(x)} = u(x)^{-1}
     \]

2. **Give the expression of \( u' \):**

   To find \( u'(x) \), we differentiate \( u(x) = (x+1)^3 \) with respect to \( x \):

   \[
   u'(x) = 3(x+1)^2
   \]

3. **Give the expression of \( f' \) as a function of \( u \) and \( u' \):**

   Since \( f(x) = u^{-1}(x) \), we apply the chain rule to find \( f'(x) \). The derivative of \( u^{-1}(x) \) is:

   \[
   f'(x) = -u^{-2}(x) \cdot u'(x)
   \]

   Substituting \( u(x) = (x+1)^3 \) and \( u'(x) = 3(x+1)^2 \), we get:

   \[
   f'(x) = -\frac{1}{(x+1)^6} \cdot 3(x+1)^2 = -\frac{3}{(x+1)^4}
   \]

4. **Calculate \( u'(x) \) and \( f'(x) \):**

   - \( u'(x) = 3(x+1)^2 \)
   - \( f'(x) = -\frac{3}{(x+1)^4} \)

Would you like a more detailed explanation of any of these steps or related concepts?

### Additional Questions:
1. How do you apply the chain rule in general when differentiating composite functions?
2. Can you explain the general method for differentiating \( u^{-1}(x) \)?
3. What is the relationship between \( f(x) \) and \( u(x) \) in this context?
4. How would the function change if we replaced \( (x+1) \) with a more complex expression?
5. Can you verify the calculations by checking the consistency of the derivative formula?

### Tip:
When differentiating functions of the form \( u^{-1}(x) \), always apply the chain rule carefully, ensuring you multiply by the derivative of the inner function \( u'(x) \).

Soit f la fonction définie sur IR par f(x) = 1 / (x + 1)^3. 1) f est de la forme u^-1 avec u = (x+1)^3. Donner l'expression de la fonction u. 2) Donner l'expression de f' en fonction de u et de u'. Calculer u'(x) puis f'(x).

Solve the problem involving the function f(x) = 1 / (x + 1)^3. Learn how to express f in terms of u, calculate u'(x), and apply the chain rule to find f'(x). Suitable for Grades 11-12.

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