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NCERT Class 12 · Ch 5 Exercise 5.5 · Q17

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 17

Differentiate (x²−5x+8)(x³+7x+9) by (i) product rule, (ii) expansion, (iii) logarithmic differentiation. Compare.

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(5 x^{4} - 20 x^{3} + 45 x^{2} - 52 x + 11\)
  1. (i) Product rule gives (2x−5)(x³+7x+9)+(x²−5x+8)(3x²+7).
  2. (ii) Expanding gives \(x^{5} - 5 x^{4} + 15 x^{3} - 26 x^{2} + 11 x + 72\). Termwise differentiation gives \(5 x^{4} - 20 x^{3} + 45 x^{2} - 52 x + 11\).
  3. (iii) Where neither factor vanishes, differentiate log|f|: f′/f=(2x−5)/(x²−5x+8)+(3x²+7)/(x³+7x+9). Multiply by f and simplify.
  4. All methods give the same polynomial derivative. The expanded formula also extends to zeros where the logarithmic intermediate is undefined.

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