Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Differentiation

NTA Abhyas 2020 · Mathematics · Differentiation

If x^ y . y^ x = 16 , then the value of (dy)/(dx) at ( 2, 2 ) is

If xy.yx=16x^{y} . y^{x} = 16, then the value of dydx\dfrac{dy}{dx} at (2,2)( 2, 2 ) is

Options

  1. A−1- 1
  2. B00
  3. C11
  4. DNone of these

Correct answer

A. −1- 1

Step-by-step solution

Since, xy.yx=16x^{y} . y^{x} = 16

∴log⁡exy+log⁡eyx=log⁡e16∴ \log_{\mathrm{e}} x^{y} + \log_{\mathrm{e}} y^{x} = \log_{\mathrm{e}} 16

⇒ylog⁡ex+xlog⁡ey=4log⁡e2⇒ y\log_{\mathrm{e}} x + x\log_{\mathrm{e}} y = 4\log_{\mathrm{e}} 2

Now, on differentiating both sides w.r.t. x,x, we get

yx+log⁡exdydx+xydydx+log⁡ey.1=0\dfrac{y}{x} + \log_{\mathrm{e}} x\dfrac{dy}{dx} + \dfrac{x}{y}\dfrac{dy}{dx} + \log_{\mathrm{e}} y . 1 = 0

⇒dydx=−(log⁡ey+yx)(log⁡ex+xy)⇒ \dfrac{dy}{dx} = - \dfrac{\left(\log_{\mathrm{e}} y +\dfrac{y}{x}\right)}{\left(\log_{\mathrm{e}} x + \dfrac{x}{y}\right)}

∴ {\left\dfrac{dy}{dx}\right|}_{( 2,2 )} = - \dfrac{( \log_{\mathrm{e}} 2 + 1 )}{( \log_{\mathrm{e}} 2 + 1 )} = - 1
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 NTA Abhyas JEE Main PYQs