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Class 12 Continuity And Differentiability · Long Answer3 Marksparametric_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #86

For x=\(a \sec^{3}{\left(t \right)}\), y=\(a \tan^{3}{\left(t \right)}\), find d²y/dx² at t=\(\frac{\pi}{4}\). Assume a≠0.

✅ Step-by-Step Solution

Final Answer: \(\frac{1}{12 a}\)
  1. Differentiate in t: dx/dt=\(3 a \tan{\left(t \right)} \sec^{3}{\left(t \right)}\) and dy/dt=\(a \left(3 \tan^{2}{\left(t \right)} + 3\right) \tan^{2}{\left(t \right)}\).
  2. Thus dy/dx=(dy/dt)/(dx/dt)=\(\sin{\left(t \right)}\).
  3. Use d²y/dx²=[d(dy/dx)/dt]/(dx/dt)=\(\frac{\cos^{4}{\left(t \right)}}{3 a \tan{\left(t \right)}}\).
  4. Therefore d²y/dx²=\(\frac{1}{12 a}\).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate in t: dx/dt=\(3 a \tan{\left(t \right)} \sec^{3}{\left(t \right)}\) and dy/dt=\(a \left(3 \tan^{2}{\left(t \right)} + 3\right) \tan^{2}{\left(t \right)}\).
  • [1 mark] Thus dy/dx=(dy/dt)/(dx/dt)=\(\sin{\left(t \right)}\).
  • [1 mark] Use d²y/dx²=[d(dy/dx)/dt]/(dx/dt)=\(\frac{\cos^{4}{\left(t \right)}}{3 a \tan{\left(t \right)}}\).
  • [0.5 marks] Therefore d²y/dx²=\(\frac{1}{12 a}\).

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