Q. 7 - Calculus

Question

Calculate the derivative of $f\left(x\right)=e^x$ at $c = 0$. At some point you should need the characterization of e given in Theorem 1.26.

Step-by-Step Solution

Answer

The derivative is $0$. 

Step 1. Given Information.

We are given a function, 

$f\left(x\right)=e^x$

Step 2. Finding the derivative.

The derivative is given by, 

$$\begin{gathered} f^{\text{'}}\left(c\right)=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ {f}^{\text{'}}\left(0\right)=\underset{h\to 0}{\lim }\frac{e^{0+h}-e^0}{h} \\ =\underset{h\to 0}{\lim }\frac{e^0·e^h-e^0}{h} \\ =\underset{h\to 0}{\lim }\frac{0}{h} \\ =0 \end{gathered}$$