Find `dy/dx if y= 1/2 -x^4`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = \frac{1}{2} - x^4 \), we will differentiate the function with respect to \( x \). Here are the steps:

### Step 1: Identify the function
We have:
\[ y = \frac{1}{2} - x^4 \]

### Step 2: Differentiate the constant term
The derivative of a constant (in this case, \( \frac{1}{2} \)) is 0. Therefore:
\[ \frac{d}{dx}\left(\frac{1}{2}\right) = 0 \]

### Step 3: Differentiate the \( -x^4 \) term
Using the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \), we differentiate \( -x^4 \):
\[ \frac{d}{dx}(-x^4) = -4x^{4-1} = -4x^3 \]

### Step 4: Combine the results
Now, we combine the results from Step 2 and Step 3:
\[ \frac{dy}{dx} = 0 - 4x^3 = -4x^3 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -4x^3 \]

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