Explained: "Find all derivatives of the function y=exx2."

alesterp

Answered question

2021-05-26

Find all derivatives of the function

y = e^{x} x^{2}

Isma Jimenez

Skilled2021-05-27Added 84 answers

Step 1
We find the first few derivatives using the product rule

y = e^{x} x^{2}

y^{(1)} = e^{x} x^{2} + 2 e^{x} x

y^{(2)} = (e^{x} x^{2} + 2 e^{x} x) + 2 e^{x} x + 2 e^{x} = e^{x} x^{2} + 4 e^{x} x + 2 e^{x}

y^{(3)} = (e^{x} x^{2} + 2 e^{x} x) + (4 e^{x} x + 4 e^{x}) + 2 e^{x} = e^{x} x^{2} + 6 e^{x} x + 6 e^{x}

y^{(4)} = (e^{x} x^{2} + 2 e^{x} x) + (6 e^{x} x + 6 e^{x}) + 6 e^{x} = e^{x} x^{2} + 8 e^{x} x + 12 e^{x}

y^{(5)} = (e^{x} x^{2} + 2 e^{x} x) + (8 e^{x} x + 8 e^{x}) + 12 e^{x} = e^{x} x^{2} + 10 e^{x} x + 20 e^{x}

Step 2
The coefficient of

e^{x} x

is a sequence 2,4,6,8,10,...
So the nth term of it will be 2n.
The coefficient of

e^{x}

is a sequence 0,2,6,12,20,...
So the nth term of it will be n(n-1)
So,

y^{(n)} = e^{x} x^{2} + 2 \neq^{x} x + n (n - 1) e^{x}

y^{(n)} = e^{x} (x^{2} + 2 n x + n^{2} - n)

Answer:

y^{(n)} = e^{x} (x^{2} + 2 n x + n^{2} - n)

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