Method of DifferentiationmediumFree

Method of Differentiation: Question

JEE Maths question with a full step-by-step solution.

Question
If f(x)=4x5+3x3+2x2+ex/7f(x) = 4x^5 + 3x^3 + 2x^2 + e^{x/7} and g(x)=f−1(x)g(x) = f^{-1}(x), then g′(1)g'(1) is.
Solution
Answer: 7
Step 1: Use the inverse function derivative formula
g[f(x)]=x  ⟹  g′[f(x)]⋅f′(x)=1g[f(x)] = x \implies g'[f(x)]\cdot f'(x) = 1
 Since f(0)=1f(0) = 1, evaluating at x=0x = 0: g′(1)⋅f′(0)=1g'(1)\cdot f'(0) = 1. Step 2: Compute f′(0)f'(0)
f′(x)=20x4+9x2+4x+ex/77  ⟹  f′(0)=17f'(x) = 20x^4 + 9x^2 + 4x + \frac{e^{x/7}}{7} \implies f'(0) = \frac{1}{7}
 Step 3: Find g′(1)g'(1)
g′(1)=1f′(0)=7g'(1) = \frac{1}{f'(0)} = 7
 Answer: 7
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