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Question

The nth derivative of h(x)=e3x+5x2 at x=0 is

A
e53n2n(n1)
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B
e53n+2n(n1)
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C
e53nn(n1)
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D
e53n2n(n+1)
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Solution

The correct option is A e53n2n(n1)
Given:
h(x)=e3x+5x2

Differentiating h(x) both sides, we get
h(x)=3e3x+5x2+2xe3x+5
h(0)=0

Again on differentiating h(x), we get
h′′(x)=3h(x)+6xe3x+5+2e3x+5
h′′(x)=3h(x)+6xe3x+5+2e3x+5
h′′(x)=3h(x)+3(h(x)h(x))+2e3x+5
h′′(x)=6h(x)3h(x)+2e3x+5
h′′(0)=2e5
Differenting h′′(x), we get
h′′′(x)=6h′′(x)3h(x)+6e3x+5
h′′′(0)=18e5
... and so on.

According to the pattern, the value of nth derivative of h(x) at x=0 is
hn(0)=e53n2n(n1)

Hence, option A.

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