Class 12 Continuity And Differentiability · Long Answer3 Markschain_and_product_rules
Class 12 Maths · Continuity And Differentiability · Long Answer #134
Let f(x)=√(x²+1), g(x)=(x+1)/(x²+1), h(x)=2x−3. Evaluate f′(h′(g′(x))).
✅ Step-by-Step Solution
Final Answer: 2/√5
- First f′(u)=u/√(u²+1), by the chain rule.
- The quotient rule gives g′(x)=[(x²+1)−2x(x+1)]/(x²+1)²=(1−2x−x²)/(x²+1)².
- For every input v, h′(v)=2. Hence h′(g′(x))=2 and f′(h′(g′(x)))=f′(2).
- Therefore the requested value is 2/√5.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] First f′(u)=u/√(u²+1), by the chain rule.
- [1 mark] The quotient rule gives g′(x)=[(x²+1)−2x(x+1)]/(x²+1)²=(1−2x−x²)/(x²+1)².
- [1 mark] For every input v, h′(v)=2. Hence h′(g′(x))=2 and f′(h′(g′(x)))=f′(2).
- [0.5 marks] Therefore the requested value is 2/√5.
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