Class 12 Continuity And Differentiability · Short Answer2 Marksparametric_derivatives
Class 12 Maths · Continuity And Differentiability · Short Answer #54
For x=\(a t^{2}\), y=\(2 a t\), find d²y/dx². Assume a≠0 and t≠0.
✅ Step-by-Step Solution
Final Answer: \(- \frac{1}{2 a t^{3}}\)
- Differentiate in t: dx/dt=\(2 a t\) and dy/dt=\(2 a\).
- Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{1}{t}\). Applying d/dx=(1/(dx/dt))d/dt gives d²y/dx²=\(- \frac{1}{2 a t^{3}}\).
- Therefore d²y/dx²=\(- \frac{1}{2 a t^{3}}\).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Differentiate in t: dx/dt=\(2 a t\) and dy/dt=\(2 a\).
- [1 mark] Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{1}{t}\). Applying d/dx=(1/(dx/dt))d/dt gives d²y/dx²=\(- \frac{1}{2 a t^{3}}\).
- [0.5 marks] Therefore d²y/dx²=\(- \frac{1}{2 a t^{3}}\).
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