If `y=(1-x)^(4)," then "y'=`

Options

A. `4(1+3x+3x^(2)+x^(3))`
B. `4(x^(3)-3x^(2)+3x-1)`
C. `-4(1+3x-3x^(2)+x^(2))`
D. `-4(x^(3)-3x^(2)+3x-1)`

Answer

To find the derivative of the function \( y = (1 - x)^4 \), 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 \( u^4 \) where \( u = (1 - x) \).

### Step 2: Differentiate the outer function
The derivative of \( u^4 \) with respect to \( u \) is:
\[ \frac{dy}{du} = 4u^3 \]

### Step 3: Differentiate the inner function
Next, we differentiate the inner function \( u = (1 - x) \):
\[ \frac{du}{dx} = -1 \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 4u^3 \cdot (-1) = -4u^3 \]

### Step 5: Substitute back the inner function
Now we substitute back \( u = (1 - x) \):
\[ \frac{dy}{dx} = -4(1 - x)^3 \]

### Step 6: Expand the expression (optional)
If required, we can expand \( (1 - x)^3 \):
\[ (1 - x)^3 = 1 - 3x + 3x^2 - x^3 \]
Thus,
\[ \frac{dy}{dx} = -4(1 - 3x + 3x^2 - x^3) = -4 + 12x - 12x^2 + 4x^3 \]

### Final Answer
So, the derivative \( y' \) is:
\[ \frac{dy}{dx} = 4x^3 - 12x^2 + 12x - 4 \]

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