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Question Number 158687 by mahdipoor last updated on 07/Nov/21

$$\mathrm{\forall }x\in Rf\left(x\right)=f{}^{\prime }\left(x\right)andf\left(0\right)=1$$$$provef(a+b)=f\left(a\right)\times f\left(b\right)$$

Answered by mr W last updated on 07/Nov/21

$$y^{\prime }=\frac{dy}{dx}=y$$$$\frac{dy}{y}=dx$$$$\int \frac{dy}{y}=\int dx$$$$\ln y-\ln C=x$$$$\ln \frac{y}{C}=x$$$$\Rightarrow y={Ce}^x$$$$y\left(0\right)=C=1$$$$\Rightarrow y=f\left(x\right)=e^x$$$$f(a+b)=e^{a+b}=e^a{e}^b=f\left(a\right)f\left(b\right)$$

Commented by mahdipoor last updated on 07/Nov/21

$$canyouprovewithoutuse{e}^x?$$

Commented by mr W last updated on 07/Nov/21

$$f\left(x\right)=e^{x}istheonlysolutionsatifying$$$$f^{\prime }\left(x\right)=f\left(x\right)andf\left(0\right)=1.$$

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