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JEE Main 20209 JanuaryEvening Shift
Exam
JEE Main
Year
2020
Date
9 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Differentiation

JEE Main 2020 · 9 January · Evening Shift · Mathematics · Differentiation

Let f and g be differentiable functions on R such that fog is the...

Let ff and gg be differentiable functions on RR such that fogfog is the identity function. If for some a,b∈R,g′(a)=5a, b \in R, {g}^{'}\left(a\right) = 5 and g(a)=b,g\left(a\right) = b, then f′(b){f}^{'}\left(b\right) is equal to:

Options

  1. A15\dfrac{1}{5}
  2. B11
  3. C55
  4. D25\dfrac{2}{5}

Correct answer

A. 15\dfrac{1}{5}

Step-by-step solution

f(g(x))=xf\left(g\left(x\right)\right) = x

⇒f′(g(x)).g′(x)=1⇒ {f}^{'}\left(g\left(x\right)\right) . {g}^{'}\left(x\right) = 1

Put x=ax = a

⇒f′(g(a))g′(a)=1⇒ {f}^{'}\left(g\left(a\right)\right){g}^{'}\left(a\right) = 1

⇒f′(b)×5=1⇒ {f}^{'}\left(b\right) \times 5 = 1

⇒f′(b)=15⇒ {f}^{'}\left(b\right) = \dfrac{1}{5}

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