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Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #101

For y=\(\tan{\left(x \right)} + \sec{\left(x \right)}\), prove \(1\)y″ + \(0\)y′ + \(0\)y = \(\frac{\cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\). Assume cos x≠0.

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Differentiate y=\(\tan{\left(x \right)} + \sec{\left(x \right)}\) to obtain y′=\(\frac{\sin{\left(x \right)} + 1}{\cos^{2}{\left(x \right)}}\).
  2. Differentiate again, using the product and chain rules: y″=\(\frac{2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 2}{\cos^{3}{\left(x \right)}}\).
  3. Substitute these derivatives into the left side \(1\)y″ + \(0\)y′ + \(0\)y = \(\frac{\cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\). Collecting like terms gives \(\frac{2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 2}{\cos^{3}{\left(x \right)}}\).
  4. This equals \(\frac{\cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\), proving the identity on the stated domain.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate y=\(\tan{\left(x \right)} + \sec{\left(x \right)}\) to obtain y′=\(\frac{\sin{\left(x \right)} + 1}{\cos^{2}{\left(x \right)}}\).
  • [1 mark] Differentiate again, using the product and chain rules: y″=\(\frac{2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 2}{\cos^{3}{\left(x \right)}}\).
  • [1 mark] Substitute these derivatives into the left side \(1\)y″ + \(0\)y′ + \(0\)y = \(\frac{\cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\). Collecting like terms gives \(\frac{2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 2}{\cos^{3}{\left(x \right)}}\).
  • [0.5 marks] This equals \(\frac{\cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\), proving the identity on the stated domain.

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