Detailed solution to Problem Set 15 Exercise 1

Question:

Find the differential of 2x² − 7x + 3.

Solutions:

Since this is a polynomial, you can easily find the derivative first; it's 2(2x²⁻¹) − 7(1) + 0, which is 4x − 7. Then multiply by dx to get (4x − 7) dx, which you can expand to 4x dx − 7 dx if you want.

Or more directly, apply the Translate Rule (d(u + C) = du) and Difference Rule (d(u − v) = du − dv) for differentials, then the Multiple Rule (d(ku) = k du), then the Power Rule (d(uⁿ) = nuⁿ⁻¹ du): d(2x² − 7x + 3) = d(2x²) − d(7x) = 2 d(x²) − 7 dx = 2(2x²⁻¹ dx) − 7 dx = 4x dx − 7 dx, which you can factor as (4x − 7) dx if you want.

Or if you want to learn fewer rules, treat 2x² − 7x + 3 as 2x² + (−7x) + 3, and use the Sum Rule (d(u + dv) = du + dv) twice, then the Product Rule (d(uv) = v du + u dv) and the Constant Rule (dK = 0), then the Power Rule. (Those four rules will cover all of the differentials in this assignment, without the need for the Translate Rule, the Difference Rule, the Multiple Rule, the Quotient Rule, or the Root Rule.) So: d(2x² + (−7x) + 3) = d(2x²) + d(−7x) + d(3) = 2 d(x²) + (−7) dx + 0 = 2(2x²⁻¹ dx) − 7 dx = 4x dx − 7 dx, which again you can factor as (4x − 7) dx if you want.

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