Question

Use the definition of the derivative and the definition of the number e to prove that $f\left(x\right)=e^x$is its own derivative.

Short answer

It is proved that derivative of the given function is$e^x$only.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=e^x$

Step 2. Calculation

As we know that, from the definition of derivative, $\frac{d}{dx}\left(b^x\right)=b^x\left(\underset{h\to 0}{lim}\frac{e^h-1}{h}=1\right)\mathrm{where}\underset{h\to 0}{lim}\frac{e^h-1}{h}=1$

Now, we will substitute $b=e$then the function becomes,

$$\begin{gathered} \frac{d}{dx}\left(e^x\right)=\underset{h\to 0}{lim}\frac{e^{x+h}-e^x}{h} \\ =e^x\left(\underset{h\to 0}{lim}\frac{e^h-1}{h}\right) \\ =e^x\left(1\right) \\ =e^x \end{gathered}$$