Differentiation of Exponential Functions

Example 1:

$$ y=e^{-x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{-x}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{-x}$$ $$ \textup{Let}\;\; u= -x$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (-x)$$ $$ \frac{dy}{dx}=e^u(-1)$$ $$ \frac{dy}{dx}=-e^u$$ $$ \frac{dy}{dx}=-e^{-x}$$