Question

The function $f\left(x\right)=e^x$is its own derivative. Are there other functions with this property? If not, explain why not. If so, give three examples

Short answer

Yes, any function with from $y=a{e}^x$has its own derivative

Three examples are :$y=e^{kx},y=c{e}^{kx},y=b^x$where $c,karecons\tan tsandb=e$

Step-by-step solution

.Given Information

We are given that the function $f\left(x\right)=e^x$has it's own derivative

. Functions with their own derivative

Any function of the form $y=a{e}^x$has its own derivative we can prove it :

Let $\frac{dy}{dx}=y$, where $y=f\left(x\right)$is an arbitrary function:-

$$\begin{gathered} \int \frac{dy}{y}=\int x \\ \ln y=x+C \\ \ln y=x+\ln a,whereC=\ln a \\ \ln y-\ln a=x \\ \ln \left(\frac{y}{a}\right)=x \\ \frac{y}{a}=e^x \\ y=a{e}^x \end{gathered}$$

. Examples

Three examples are :

$y=e^{kx},y=c{e}^{kx}y=b^x$where$candkarecons\tan tsandb=e$