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Question

Let f(x) be a polynomial in x. Then the second derivation of f(ex), is:

A
f"(ex).ex+f(ex)
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B
f"(ex).ex+f(ex).e2x
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C
f"(ex)e2x
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D
f"(ex)e2x+f(ex).ex
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Solution

The correct option is D f"(ex)e2x+f(ex).ex

We have,

f(x) is a polynomial.

Then,

The second order of f(ex)=?

Find that,

ddx[ddxf(ex)]=?

So,

We know that,

ddx(ex)=ex

So,

ddx[f(g(x))]=g(x)f(g(x))

Now. Using formula,

ddx(I.II)=I.ddxII+II.ddxI

So,

ddx[f(ex)]=ex.f(ex)......(1)

Again differentiating and we get,

ddx[ddxf(ex)]=ddx[f(ex).ex]=f(ex)ex+ex[exf′′(ex)]

=ddx[ddxf(ex)]=exf(ex)+e2xf′′(ex)

Hence, this is the answer.


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