Q. 6 - Calculus

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

Step-by-Step Solution

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 $c,k are cons\tan ts and b=e$

Step 1 .Given Information

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

Step 2 . 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:-

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

Step 2 . Examples

Three examples are :

$y=e^{kx},y=c{e}^{kx}y=b^x$ where $c and k are cons\tan ts and b=e$