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

Options

A. 1
B. `-1`
C. x
D. `sqrtx`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = (1 + x^{1/4})(1 + x^{1/2})(1 - x^{1/4})\), we will follow these steps:

### Step 1: Rewrite the expression for \(y\)
We can rewrite \(y\) as:
\[ y = (1 + x^{1/4})(1 - x^{1/4})(1 + x^{1/2}) \]
Using the difference of squares, we can simplify the first two terms:
\[ (1 + x^{1/4})(1 - x^{1/4}) = 1 - (x^{1/4})^2 = 1 - x^{1/2} \]
Thus, we can rewrite \(y\) as:
\[ y = (1 - x^{1/2})(1 + x^{1/2}) \]

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

### Step 3: Differentiate \(y\) with respect to \(x\)
Now we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(1 - x) = 0 - 1 = -1 \]

### Final Answer
Thus, we have:
\[ \frac{dy}{dx} = -1 \]