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BITSAT2023MathematicsDifferentiationActual

If y^x=e^ y-x , then d y d x is equal to

Options

  1. A1+ y y y
  2. B(1+ y)^2 y y
  3. C1+ y ( y)^2
  4. D(1+ y)^2 y

Correct answer

D. (1+ y)^2 y

Step-by-step solution

Here, y ^ x = e ^ y - x Taking log on both sides, we get aligned & y^x= e^ y-x & ( a^b=b a and e=1 ) aligned x y=(y-x) e x y=y-x (i) On differentiating w.r.t. x , we get d d x (x y)= d d x (y-x) (using product rule) aligned & x ( 1 y ) d y d x + y(1)= d y d x -1 & d y d x ( x y -1 )=-1- y & d y d x [ y (1+ y) y -1 ]=-(1+ y) & [ from eq. (i) x= y (1+ y) ] & d y d x [ 1-1- y 1+ y ]=-(1+ y) aligned d y d x =- (1+ y)^2 - y d y d x = (1+ y)^2 y

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