Detailed solution to Problem Set 15 Exercise 7

Question:

Find d(4 + 4/x).

Solutions:

You can use the Quotient Rule (d(u/v) = (v du − u dv)/v²): d(4 + 4/x) = d(4/x) = (x d(4) − 4 dx)/x² = (x ⋅ 0 − 4 dx)/x² = −4 dx/x², which you can write as (−4/x²) dx or −4x⁻² dx if you want.

Or you can avoid the Quotient Rule by treating 4/x as 4x⁻¹: d(4 + 4x⁻¹) = d(4x⁻¹) = 4 d(x⁻¹) = 4(−1x⁻¹⁻¹ dx) = −4x⁻² dx, which you can write as (−4/x²) dx or −4 dx/x² if you want.

Or you can use the relatively obscure Reciprocal Rule (d(/u) = −du/u², where /a indicates the reciprocal of a so that b/a can be read as b ⋅ /a): d(4 + 4/x) = d(4/x) = 4 d(/x) = 4(−dx/x²) = −4 dx/x², which again you can write as (−4/x²) dx or −4x⁻² dx if you want.

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