If `y=(1+x^(1//4))(1+x^(1//2))(1-x^(1//4))`, then what is `(dy)/(dx)` equal to ?

Options

A. A) 1
B. B) -1
C. C) 0
D. D) -2x

Answer

To find the derivative of the function \( y = (1 + x^{1/4})(1 + x^{1/2})(1 - x^{1/4}) \), we can simplify the expression first and then differentiate.

### Step-by-step Solution:

1. **Rewrite the function**:
\[ y = (1 + x^{1/4})(1 + x^{1/2})(1 - x^{1/4}) \]

2. **Use the difference of squares**:
Notice that \( (1 + x^{1/4})(1 - x^{1/4}) \) can be simplified using the formula \( (a+b)(a-b) = a^2 - b^2 \):
\[ (1 + x^{1/4})(1 - x^{1/4}) = 1^2 - (x^{1/4})^2 = 1 - x^{1/2} \]
Thus, we can rewrite \( y \) as:
\[ y = (1 - x^{1/2})(1 + x^{1/2}) \]

3. **Apply the difference of squares again**:
Now, we can apply the difference of squares again:
\[ y = (1 - x^{1/2})(1 + x^{1/2}) = 1^2 - (x^{1/2})^2 = 1 - x \]

4. **Differentiate \( y \)**:
Now that we have simplified \( y \) to \( 1 - x \), we can differentiate it:
\[ \frac{dy}{dx} = \frac{d}{dx}(1 - x) \]
The derivative of \( 1 \) is \( 0 \) and the derivative of \( -x \) is \( -1 \):
\[ \frac{dy}{dx} = 0 - 1 = -1 \]

### Final Answer:
\[ \frac{dy}{dx} = -1 \]

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