Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2018 · Mathematics · Continuity and Differentiability

Let f(x+y)=f(x) f(y) and f(x)=1+x g(x) φ(x) , where _ x arrow 0...

Let f(x+y)=f(x)f(y)f(x+y)=f(x) f(y) and f(x)=1+xg(x)ϕ(x)f(x)=1+x g(x) \phi(x), where lim⁡x→0g(x)=a\lim _{x \rightarrow 0} g(x)=a and lim⁡x→0ϕ(x)=b\lim _{x \rightarrow 0} \phi(x)=b. Then what is f′(x)f^{\prime}(x) equal to?

Options

  1. A1+abf(x)1+\mathrm{abf}(\mathrm{x})
  2. B1+ab1+\mathrm{ab}
  3. Cab
  4. Dabf⁡(x)\operatorname{abf}(\mathrm{x})

Correct answer

D. abf⁡(x)\operatorname{abf}(\mathrm{x})

Step-by-step solution

f′(x)lim⁡h→0f(x+h)−f(x)h=lim⁡h→0f(x)f(h)−f(x)h=lim⁡h→0f(x)[f(h)−1]h=lim⁡h→0f(x)[hg(h)ϕ(h)]h=abf(x)\begin{aligned} & f^{\prime}(x) \\ & \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{f(x) f(h)-f(x)}{h} \\ & =\lim _{h \rightarrow 0} \frac{f(x)[f(h)-1]}{h}=\lim _{h \rightarrow 0} \frac{f(x)[h g(h) \phi(h)]}{h} \\ & =a b f(x)\end{aligned}
Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

What is the derivative of (x)/(|x|) with respect to x , where x< 0 ? NDA 2026Passage: Let f(x)= cases ax(x-1), & x 3 cases Given that f(x) is continuous for all x but… NDA 2026Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If q(x)=f ^ ∘ g(x) , then which… NDA 2026Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If p(x)=f(x)g(x) , then which of… NDA 2026If f(x) is differentiable at x=a , then consider the following statements : I. f(x) is… NDA 2026Passage: Let f(x)= cases ax(x-1), & x 3 cases Given that f(x) is continuous for all x but… NDA 2026Consider the following for the two (02) items that follow: Let f(x) = cases x^3, & x^2… NDA 2025For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on… NDA 2025For the following two (02) items: Consider the function f(x) = cases 4(5^x), & x If the… NDA 2025For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on… NDA 2025Consider the following for the two (02) items that follow: Let f(x) = cases x^3, & x^2… NDA 2025Paragraph: Let f(x)=[x]^2- [x^2] . Question: Consider the following statements: A. f(x)… NDA 2024Let f(x)=|x|+1 and g(x)=[x]-1 , where [.] is the greatest integer function. Let… NDA 2024Let f(x)=\ array cl (x-3)/(|x-3|)+a ; & x 3 array .f(x) be continuous at x=3 . What is… NDA 2023Let f(x)=\ array cl (x-3)/(|x-3|)+a ; & x 3 array .f(x) be continuous at x=3 . What is… NDA 2023Let f(x)=\ array cc a x(x+1)+b, & x < 1 x-1 & 1 ≤ x ≤ 2 array . If the function f(x) is… NDA 2023 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs