More practice questions

  1. Q5. Differentiate y = √(P/Q), where P = (x − 1)(x − 2) and Q = (x − 3)(x − 4)(x − 5), for x > 5 (NCERT Exercise 5.5, Q2, page 134).

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    log y = ½(log P − log Q), and log P and log Q split into sums of logs of the factors. So y′/y = ½(A − B), where A is the sum of 1/(x − 1) and 1/(x − 2), and B is the sum of 1/(x − 3), 1/(x − 4) and 1/(x − 5); the half applies to both. Multiply by y for y′.

  2. Q6. Differentiate (x + 3)2(x + 4)3(x + 5)4 (NCERT Exercise 5.5, Q5, page 134).

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    y′/y = 2/(x + 3) + 3/(x + 4) + 4/(x + 5). Over the common denominator (x + 3)(x + 4)(x + 5), the top is Q = 9x2 + 70x + 133. So y′ is Q times (x + 3)(x + 4)2(x + 5)3.

  3. Q7. Differentiate xsin x + (sin x)cos x, for 0 < x < π (NCERT Exercise 5.5, Q9, page 134).

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    For u = xsin x, u′ = u(cos x log x + sin x/x), as in NCERT Example 29. For v = (sin x)cos x, log v = cos x log sin x, so v′ = v(A − B), with A = cos x cot x and B = sin x log sin x. The derivative is u′ + v′.

  4. Q8. Find dy/dx if (cos x)y = (cos y)x (NCERT Exercise 5.5, Q14, page 134).

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    For cos x > 0 and cos y > 0, y log cos x = x log cos y. Differentiating: y′ log cos x − y tan x = log cos y − x tan y y′. So y′(log cos x + x tan y) = log cos y + y tan x, and y′ = (y tan x + log cos y)/(x tan y + log cos x).

  5. Q9. Show by logarithms that the derivative of u·v·w is u′vw + uv′w + uvw′ (NCERT Exercise 5.5, Q18, page 134).

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    For positive u, v, w: log (uvw) = log u + log v + log w, so (uvw)′/(uvw) = u′/u + v′/v + w′/w. Multiplying by uvw gives u′vw + uv′w + uvw′. (Repeated use of the product rule gives the same, and that route works for all signs.)

  6. Q10. Differentiate xx2, for x > 0.

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    log y = x2 log x, so y′/y = 2x log x + x2/x = x(2 log x + 1). So y′ = xx2 + 1(2 log x + 1).