Question

find nth derivative of x^2e^x​

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

The $n^{th}$ derivative of the given function

  $y = x^2 e^x$ is given by

$y_{n} = 2(n-1)e^x + 2xe^x + y_{n-1}$

• Now we have

            $y = x^2 e^x$

• differentiating with respect to 'x', we get

       $y_{1} = 2xe^x +x^2e^x$    

      here $y = x^2 e^x$ therefore,

        $y_{1} = 2xe^x +y$      - (1)

• Now, differentiating (1) with respect to'x',we get

       $y_{2} = 2e^x +2xe^x + y_{1}$       - (2)

• differentiating (2) with respect to 'x', we get

         $y_{3} = 4e^x +2xe^x + y_{2}$       - (3)

• differentiating (3) with respect to 'x', we get

          $y_{4} = 6e^x +2xe^x + y_{3}$      - (4)

• Now (1), (2), (3) and (4) can also be written as

    $y_{1} = 2(1-1)e^x +2xe^x +y_{(1-1)}$

    $y_{2} = 2(2-1)e^x +2xe^x + y_{(2-1)}$

    $y_{3} = 2(3-1)e^x +2xe^x + y_{(3-1)}$

    $y_{4} = 2(4-1)e^x +2xe^x + y_{(4-1)}$

    Similarly we can write nth derivative of y as

      $y_{n} = 2(n-1)e^x + 2xe^x + y_{n-1}$