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AP EAMCET202421 May 2024Morning ShiftMathematicsDifferentiationActual

The rate of change of x^ x with respect to ( x)^ x is

Options

  1. Ax^ x ( x x + x x ) ( x)^x(x x+ x)
  2. Bx^ x (x x+ x) x^ x ( x x + x x )
  3. Cy ( x x + x x )
  4. D( x)^ x (x x+ x)

Correct answer

A. x^ x ( x x + x x ) ( x)^x(x x+ x)

Step-by-step solution

Let u =x^ x u = x x^ Differentiating w.r.t. x aligned & 1 u d u d x = x x+ x x & d u d x =x^ x [ x x+ x x ] aligned Let v =( x)^x v =x x Differentiating w.r.t. x aligned & 1 v d v d x = x+x x & d v d x =( x)^x[x x+ x] aligned Now, d u d v = d u d x d v d x = x^ x [ x x+ x x ] ( x)^x[x x+ x]

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