Find the derivatives of the following at any point of their domains :
`f(x)=(x+1)^(3)`

Answer

To find the derivative of the function \( f(x) = (x + 1)^3 \), we will use the chain rule of differentiation. Here are the steps to derive it:

### Step 1: Identify the function and the outer function
We have:
\[ f(x) = (x + 1)^3 \]
Let \( g(x) = x + 1 \) (the inner function) and \( h(u) = u^3 \) (the outer function where \( u = g(x) \)).

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( h(u) = u^3 \) is:
\[ h'(u) = 3u^2 \]

### Step 3: Differentiate the inner function
The derivative of the inner function \( g(x) = x + 1 \) is:
\[ g'(x) = 1 \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ f'(x) = h'(g(x)) \cdot g'(x) \]
Substituting the derivatives we found:
\[ f'(x) = 3(g(x))^2 \cdot g'(x) \]
\[ f'(x) = 3(x + 1)^2 \cdot 1 \]

### Step 5: Simplify the expression
Thus, the derivative is:
\[ f'(x) = 3(x + 1)^2 \]

### Final Answer:
The derivative of the function \( f(x) = (x + 1)^3 \) is:
\[ f'(x) = 3(x + 1)^2 \]

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