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KCET2021MathematicsDifferentiation

For constant a, d d x (x^ x +x^ a +a^ x +a^ a ) is

Options

  1. Ax^ x (1+ x)+a x^ a-1
  2. Bx^ x (1+ x)+a x^ a-1 +a^ x a
  3. Cx^ x (1+ x)+a^ a (1+ x)
  4. Dx^ x (1+ x)+a^ a (1+ a)+a x^ a-1

Correct answer

B. x^ x (1+ x)+a x^ a-1 +a^ x a

Step-by-step solution

d d x (x^ x +x^ a +a^ x +a^ a ) aligned &= d d x (x^ x )+ d d x (x^ a )+ d d x (a^ x )+ d d x (a^ a ) &= d d x (y)+a x^ a-1 +a^ x a+0...(i) aligned [ let x^ x =y ]y=x^ x Taking log on both sides, y=x x On differentiating w.r.t. x , we get aligned 1 y d y d x &=x 1 x + x d y d x &=x^ x (1+ x) aligned Substitute the value of d y d x in Eq. (i), aligned & d d x (x^ x +x^ a +a^ x +a^ a ) &=x^ x (1+ x)+a x^ a-1 +a^ x a aligned

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