If `f(1) = 3, f' (1) = -1/3`, then the derivative of `{x^11 + f (x)}^-2` at `x = 1,` is

Options

A. `-(1)/(2)`
B. -1
C. 1
D. `f'(1)`

Explanation

To find the derivative of \( (x^{11} + f(x))^{-2} \) at \( x = 1 \), we will use the chain rule and the product rule of differentiation. Let's go through the steps:

### Step 1: Define the function
Let \( y = (x^{11} + f(x))^{-2} \).

### Step 2: Differentiate using the chain rule
Using the chain rule, we differentiate \( y \):
\[ \frac{dy}{dx} = -2(x^{11} + f(x))^{-3} \cdot \frac{d}{dx}(x^{11} + f(x)) \]

### Step 3: Differentiate the inner function
Now, we need to differentiate \( x^{11} + f(x) \):
\[ \frac{d}{dx}(x^{11} + f(x)) = 11x^{10} + f'(x) \]

### Step 4: Substitute back into the derivative
Substituting this back into the derivative, we have:
\[ \frac{dy}{dx} = -2(x^{11} + f(x))^{-3} \cdot (11x^{10} + f'(x)) \]

### Step 5: Evaluate at \( x = 1 \)
Now, we need to evaluate this derivative at \( x = 1 \). We know that \( f(1) = 3 \) and \( f'(1) = -\frac{1}{3} \).

First, calculate \( x^{11} + f(x) \) at \( x = 1 \):
\[ 1^{11} + f(1) = 1 + 3 = 4 \]

Now substitute \( x = 1 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=1} = -2(4)^{-3} \cdot (11 \cdot 1^{10} + f'(1)) \]

Calculating \( 11 \cdot 1^{10} + f'(1) \):
\[ 11 \cdot 1 + f'(1) = 11 - \frac{1}{3} = \frac{33}{3} - \frac{1}{3} = \frac{32}{3} \]

### Step 6: Substitute values into the derivative
Now we can substitute everything back:
\[ \frac{dy}{dx} \bigg|_{x=1} = -2 \cdot \frac{1}{64} \cdot \frac{32}{3} \]

### Step 7: Simplify the expression
Calculating this gives:
\[ \frac{dy}{dx} \bigg|_{x=1} = -\frac{2 \cdot 32}{64 \cdot 3} = -\frac{64}{64 \cdot 3} = -\frac{1}{3} \]

Thus, the derivative of \( (x^{11} + f(x))^{-2} \) at \( x = 1 \) is:
\[ \boxed{-\frac{1}{3}} \]