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Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #99

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

✅ Step-by-Step Solution

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

📋 CBSE Stepwise Marking Scheme

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

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