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Let f(x+y)=f(x)dotf(y) for all x and y. ...

Let `f(x+y)=f(x)dotf(y)` for all `x` and `y`. Suppose `f(5)=2` and `f^(prime)(0)=3`. Find `f^(prime)(5)`.

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Verified by Experts

The correct Answer is:
6
`f(x+y)=f(x)f(y)" (1)"`
`f'(5)=underset(hrarr0)lim(f(5+h)-f(5))/(h)`
`=underset(hrarr0)lim(f(5)f(h)-f(5))/(h)`
`=f(5)underset(hrarr0)lim(f(h)-1)/(h)`
`=f(5)underset(hrarr0)lim(f(h)-1)/(h)`
In (1), replace x by 5 and y by 0. Then, `f(5+0)=f(5)cdotf(0)`
`"or "f(0)=1`
`"or "f'(5)=f(5)underset(hrarr0)lim(f(h)-f(0))/(h)`
`=f(5)f'(0)=2xx3=6`
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