Differentiate y = log(x + √(x² + a²)) with respect to x.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. Given: y = log(x + √(x² + a²)), where a is a constant. To find: dy/dx.
2 · Write the rules. d/dx log u = u′/u, and d/dx √v = v′/(2√v).
3 · Differentiate the inside. u = x + √(x² + a²), so u′ = 1 + x/√(x² + a²) = (√(x² + a²) + x)/√(x² + a²).
4 · Divide by u. dy/dx = (√(x² + a²) + x)/[√(x² + a²) × (x + √(x² + a²))].
5 · Simplify. The factor x + √(x² + a²) cancels, so dy/dx = 1/√(x² + a²).
6 · Check and box the answer. Put a = 1 to get the known result 1/√(x² + 1). Answer: dy/dx = 1/√(x² + a²).

Scoring tip: Write u′ as one fraction before dividing. The cancellation is then easy to see.