Understanding the Function
The given function is y = 1 + 3x - x². To find dy/dx, we need to differentiate this function with respect to x.
Step-by-Step Differentiation
1. Identify the Terms
- The function has three terms: 1, 3x, and -x².
2. Differentiate Each Term
- The derivative of a constant (1) is 0.
- The derivative of 3x is 3.
- The derivative of -x² is -2x.
3. Combine the Derivatives
- Therefore, dy/dx = 0 + 3 - 2x.
Final Result
After combining, we get:
- dy/dx = 3 - 2x.
Interpretation of the Result
- The derivative, dy/dx, represents the slope of the tangent line to the curve at any point x.
- A positive slope (dy/dx > 0) indicates the function is increasing, while a negative slope (dy/dx < 0)="" indicates="" it="" is="" />
Conclusion
- The expression dy/dx = 3 - 2x is a linear function of x, showing how the rate of change of y varies with x.
- This can be further analyzed to find critical points or to sketch the graph of the function.
Using this method provides a clear understanding of the differentiation process and the behavior of the function.

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