Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Differentiation question

2020 · 9 Jan · Shift 2 · Q39
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Differentiation
  5. /2020 · 9 Jan · Shift 2 · Q39

Differentiation question

2020 · 9 Jan · Shift 2 · Q39

JEE MainMathematicsDifferentiationMCQ+4 / −1
Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b ∈\in∈ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is equal to :
  1. A
    1
  2. B
    5
  3. C
    25{2 \over 5}52​
  4. D
    15{1 \over 5}51​
View written solutionFree

Correct answer: D

  1. We are given that f∘gf \circ gf∘g is the identity function on R\mathbb{R}R. That means f(g(x))=xfor all x∈R.f(g(x)) = x \quad \text{for all } x \in \mathbb{R}.f(g(x))=xfor all x∈R.

  2. Differentiate both sides with respect to xxx using the chain rule: ddx[f(g(x))]=ddx[x].\frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x].dxd​[f(g(x))]=dxd​[x]. Hence, f′(g(x)) g′(x)=1.f'(g(x))\,g'(x) = 1.f′(g(x))g′(x)=1.

  3. Now use the given values: g(a)=b,g′(a)=5.g(a) = b, \qquad g'(a) = 5.g(a)=b,g′(a)=5. Substituting x=ax=ax=a into the differentiated equation, f′(g(a)) g′(a)=1.f'(g(a))\,g'(a) = 1.f′(g(a))g′(a)=1. So, f′(b)⋅5=1.f'(b) \cdot 5 = 1.f′(b)⋅5=1.

  4. Therefore, f′(b)=15.f'(b) = \frac{1}{5}.f′(b)=51​.

  5. Checking options:

    • A: 111
    • B: 555
    • C: 25\frac{2}{5}52​
    • D: 15\frac{1}{5}51​

    So the correct option is D.

PreviousNext

More from Differentiation

  • If 2y=(cot−1(cosx−3​sinx3​cosx+sinx​))2, x ∈ (0,2π​) then dxdy​ is equal to:2019 · MCQ
  • If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :2019 · MCQ
  • If x = 3 tan t and y = 3 sec t, then the value of dx2d2y​ at t =4π​, is :2019 · MCQ
  • Let f(x) = loge(sin x), (0 < x < π) and g(x) = sin–1 (e–x ), (x ≥ 0). If α is a positive real number such that a = (fog)'(α) and b = (fog)(α), then :2019 · MCQ
  • Let f : R → R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x ∈ R. Then f(2) equals -2019 · MCQ
  • If xloge(logex) − x2 + y2 = 4(y > 0), then dxdy​ at x = e is equal to :2019 · MCQ
  • If ey + xy = e, the ordered pair (dxdy​,dx2d2y​) at x = 0 is equal to :2019 · MCQ
  • The derivative of tan−1(sinx+cosxsinx−cosx​), with respect to 2x​, where (x∈(0,2π​)) is :2019 · MCQ