1. **Problem:** Find the derivative of the function $y = 3x^2 - 2$ at $x=1$. 2. **Formula:** The derivative of $y = ax^n$ is $y' = a n x^{n-1}$. 3. **Step 1:** Differentiate $y = 3x^2 - 2$. $$y' = 3 \times 2 x^{2-1} - 0 = 6x$$ 4. **Step 2:** Evaluate the derivative at $x=1$. $$f'(1) = 6 \times 1 = 6$$ 5. **Answer:** The derivative of $y = 3x^2 - 2$ at $x=1$ is $6$. **Different notations of derivatives:** - Leibniz notation: $\frac{dy}{dx}$ - Lagrange's notation: $f'(x)$ - Newton's notation (for time derivatives): $\dot{y}$ - Euler's notation: $D_x y$ These notations all represent the rate of change of a function with respect to a variable.