Find `dy/dx` if `y=2x^7`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = 2x^7\), we will use the power rule of differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the function**: We have \(y = 2x^7\).

2. **Recall the power rule**: The power rule states that if \(y = x^n\), then \(\frac{dy}{dx} = n \cdot x^{n-1}\), where \(n\) is a constant.

3. **Differentiate the function**: Since our function has a constant multiplier (2), we can factor it out. Thus,
\[ \frac{dy}{dx} = \frac{d}{dx}(2x^7) = 2 \cdot \frac{d}{dx}(x^7) \]

4. **Apply the power rule**: Now, applying the power rule to \(x^7\):
\[ \frac{d}{dx}(x^7) = 7x^{7-1} = 7x^6 \]

5. **Combine the results**: Now substitute back into the equation:
\[ \frac{dy}{dx} = 2 \cdot 7x^6 = 14x^6 \]

6. **Final answer**: Therefore, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 14x^6 \]