Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Differentiation

JEE Main · Mathematics · Differentiation

(d)/(d x) ((x+5)/((x+1)^2(x+2)))=

\frac{d}{d x}(x+5(x+1)2(x+2))=\left(\frac{x+5}{(x+1)^2(x+2)}\right)=

Options

  1. A8(x+2)2−3(x+1)2+3(x+1)3\frac{8}{(x+2)^2}-\frac{3}{(x+1)^2}+\frac{3}{(x+1)^3}
  2. B3(x+1)2−3(x+2)2−8(x+1)3\frac{3}{(x+1)^2}-\frac{3}{(x+2)^2}-\frac{8}{(x+1)^3}
  3. C3(x+2)2−3(x+1)3−8(x+1)2\frac{3}{(x+2)^2}-\frac{3}{(x+1)^3}-\frac{8}{(x+1)^2}
  4. D8(x+2)2−3(x+1)3+3(x+1)2\frac{8}{(x+2)^2}-\frac{3}{(x+1)^3}+\frac{3}{(x+1)^2}

Correct answer

B. 3(x+1)2−3(x+2)2−8(x+1)3\frac{3}{(x+1)^2}-\frac{3}{(x+2)^2}-\frac{8}{(x+1)^3}

Step-by-step solution

Let x+5(x+1)2(x+4)=A(x+1)+B(x+1)2+C(x+4)\frac{x+5}{(x+1)^2(x+4)}=\frac{A}{(x+1)}+\frac{B}{(x+1)^2}+\frac{C}{(x+4)}⇒x+5=A(x+1)(x+2)+B(x+2)+c(x+1)2⇒x+5=x2(A+C)+x(3A+B+2C)+2A+2B+C.\begin{aligned}& \Rightarrow x+5=A(x+1)(x+2)+B(x+2)+c(x+1)^2 \\ & \Rightarrow x+5=x²(A+C)+x(3 A+B+2 C)+2 A+2 B+C . \end{aligned}OncomparingthecoefficientsoflikepowerofOn comparing the coefficients of like power ofx,weget, we getA+C=0,3A+B+2C=1 and 2A+2B+C=5\begin{aligned}A+C & =0, \\ 3 A+B+2 C & =1 \text { and }2 A+2 B+C=5 \end{aligned}OnsubstitutingOn substitutingA=-Cinlasttwoequations,wegetin last two equations, we getB−C=12B−C=5\begin{aligned}B-C & =1 \\ 2 B-C & =5 \end{aligned}and2B−C=5OnsubtractingEqs.(ii)and(iii),wegetB=4⇒C=3[fromEq.(ii)]andhenceand 2 B-C=5 On subtracting Eqs. (ii) and (iii), we get B=4 \Rightarrow C=3 [from Eq. (ii)] and henceA=-3Now,Now,\frac{(x+5)}{(x+1)^2(x+2)}=\frac{-3}{(x+1)}+\frac{4}{(x+1)^2}+\frac{3}{(x+2)}Ondifferentiatingbothsidesw.r.tOn differentiating both sides w.r.tx,weget., we get.\frac{d}{d x}\left(\frac{(x+5)}{(x+1)^2(x+4)}\right)=\frac{3}{(x+1)^2}-\frac{8}{(x+1)^3}-\frac{3}{(x+2)^2}
Practice Differentiation on Quantrex Academy →

More from Differentiation

Let ℝ denote the set of all real numbers. Consider the polynomial function f: ℝ arrow ℝ… JEE Advanced 2026Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x… JEE Main 2026Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x) , for all x ∈ ℝ . If… JEE Main 2026Let f(x)=x^ 3 +x^ 2 f^ (1)+2 x f^ (2)+f^ (3), x ∈ R . Then the value of f^ (5) is: JEE Main 2026Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ and g: ℝ arrow(0,4) be… JEE Advanced 2025Let f: R arrow R be a twice differentiable function such that ( x y)(f(2 x+2 y)-f(2 x-2… JEE Main 2025If _e y=3 ^ -1 x , then (1-x^2) y^ -x y^ at x=(1)/(2) is equal to JEE Main 2024Let f(x)=a x^3+b x^2+c x+41 be such that f(1)=40, f^ (1)=2 and f^ (1)=4 . Then… JEE Main 2024Suppose for a differentiable function h, h(0)=0, h(1)=1 and h^ (0)=h^ (1)=2 . If… JEE Main 2024If f(x)=\ array l x^3 ((1)/(x)), x ≠ 0 0 , x=0 array . then JEE Main 2024If y(θ)=(2 θ+ 2 θ)/( 3 θ+4 2 θ+5 θ+2) , then at θ=(π)/(2), y^ +y^ +y is equal to : JEE Main 2024Let f(x)=x^5+2 e^ x / 4 for all x ∈ R . Consider a function g(x) such that (g ∘ f)(x)=x… JEE Main 2024If y = ( x + 1)( x ^ 2 - x ) x x + x + x + (1)/(15)(3 ^ 2 x - 5) ^ 3 x, then 96 y ^ '… JEE Main 2024Let y = _ e ( 1 - x ^ 2 1 + x ^ 2 ), - 1 < x < 1 . Then at x = (1)/(2) , the value of… JEE Main 2024Suppose f(x) = ( 2 ^ x + 2 ^ - x ) x ^ - 1 ( x ^ 2 - x + 1) (7 x ^ 2 + 3x + 1) ^ 3 . Then… JEE Main 2024Let f ( x ) = x ^ 3 + x ^ 2 f ^ ' ( 1 ) + x f ^ " ( 2 ) + f ^ ''' ( 3 ) , x ∈ R . Then f… JEE Main 2024 Full Differentiation list Differentiation JEE Maths PYQsJEE Mathematics PYQs All 99 Percentile Qs Bank for JEE Main PYQs