Find the second order derivative if `y= x^3`

Answer

To find the second order derivative of the function \( y = x^3 \), we will follow these steps:

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = x^3 \]

Using the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \):

\[ \frac{dy}{dx} = 3 \cdot x^{3-1} = 3x^2 \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now, we differentiate \( \frac{dy}{dx} = 3x^2 \) again with respect to \( x \):

Using the power rule again:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x \]

### Final Result

Thus, the second order derivative of \( y = x^3 \) is:
\[ \frac{d^2y}{dx^2} = 6x \]

---