If x = t + 1/t and y = t − 1/t, find dy/dx and show that y (dy/dx) = x.

1. 1 · Write what is given and what is to be found. Given: x = t + 1/t and y = t − 1/t. To find: dy/dx, then show y dy/dx = x.
2. 2 · Write the formula. dy/dx = (dy/dt) ÷ (dx/dt).
3. 3 · Differentiate with respect to t.
   • dx/dt = 1 − 1/t² = (t² − 1)/t²
   • dy/dt = 1 + 1/t² = (t² + 1)/t²
4. 4 · Divide. dy/dx = (t² + 1)/(t² − 1).
5. 5 · Find y dy/dx. y dy/dx = [(t² − 1)/t] × [(t² + 1)/(t² − 1)] = (t² + 1)/t = t + 1/t = x.
6. 6 · Check and box the answer. Hence y dy/dx = x. As a check, x² − y² = 4, and implicit differentiation gives the same result.

Scoring tip: Write each derivative over t² before dividing. The t² then cancels cleanly.