Find the derivatives of the following at any point of their domains :
`f(x)=(x^(2)+2)^(2)`

Answer

To find the derivative of the function \( f(x) = (x^2 + 2)^2 \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
We can identify the outer function as \( g(u) = u^2 \) where \( u = x^2 + 2 \) is the inner function.

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

### Step 3: Differentiate the inner function
Next, we differentiate the inner function \( u = x^2 + 2 \):
\[ u' = \frac{d}{dx}(x^2 + 2) = 2x \]

### Step 4: Apply the chain rule
According to the chain rule, the derivative of \( f(x) \) is given by:
\[ f'(x) = g'(u) \cdot u' \]
Substituting \( g'(u) \) and \( u' \):
\[ f'(x) = 2(x^2 + 2) \cdot (2x) \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ f'(x) = 4x(x^2 + 2) \]

### Final Answer
Thus, the derivative of the function \( f(x) = (x^2 + 2)^2 \) is:
\[ f'(x) = 4x(x^2 + 2) \]

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