The derivative of `f(x) = (x - 4)^(2)` is

Options

A. 2(x - 2)
B. `x^(2) - 4`
C. x - 4
D. 2(x - 4)

Answer

To find the derivative of the function \( f(x) = (x - 4)^2 \), we will use the power rule of differentiation. Here are the steps:

### Step 1: Identify the function
The function given is:
\[ f(x) = (x - 4)^2 \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we can use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[ g'(h(x)) \cdot h'(x) \]
In our case, let \( g(u) = u^2 \) where \( u = (x - 4) \).

### Step 3: Differentiate the outer function
The derivative of \( g(u) = u^2 \) is:
\[ g'(u) = 2u \]

### Step 4: Differentiate the inner function
Now we differentiate the inner function \( h(x) = x - 4 \):
\[ h'(x) = 1 \]

### Step 5: Combine the derivatives
Now, applying the chain rule:
\[ f'(x) = g'(h(x)) \cdot h'(x) = 2(x - 4) \cdot 1 \]

### Step 6: Write the final answer
Thus, the derivative of the function is:
\[ f'(x) = 2(x - 4) \]

### Summary
The derivative of \( f(x) = (x - 4)^2 \) is:
\[ f'(x) = 2(x - 4) \]

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