Find `dy/dx if y=4x-x^3`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = 4x - x^3\), we will use the rules of differentiation. Let's go through the steps one by one.

### Step 1: Identify the function
We start with the function:
\[ y = 4x - x^3 \]

### Step 2: Differentiate each term
We will differentiate each term of the function separately.

1. **Differentiate \(4x\)**:
- The derivative of \(4x\) with respect to \(x\) is simply \(4\) because the derivative of \(x\) is \(1\) and \(4\) is a constant.

2. **Differentiate \(-x^3\)**:
- Using the power rule for differentiation, which states that if \(y = x^n\), then \(\frac{dy}{dx} = nx^{n-1}\).
- Here, \(n = 3\), so the derivative of \(-x^3\) is \(-3x^{3-1} = -3x^2\).

### Step 3: Combine the derivatives
Now, we combine the derivatives of both terms:
\[ \frac{dy}{dx} = 4 - 3x^2 \]

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

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