Question

find the difference quotient $\frac{f(x + h)-f(x)}{h}$. write the answer in factored form.

$f(x)=e^{x}$

the difference quotient for the equation $f(x)=e^{x}$ is

Explanation:

Step1: Substitute into the difference quotient formula

Given $f(x)=e^{x}$, then $f(x + h)=e^{x + h}$.
The difference quotient $\frac{f(x + h)-f(x)}{h}=\frac{e^{x + h}-e^{x}}{h}$.

Step2: Use the exponent rule $a^{m + n}=a^{m}\cdot a^{n}$

Since $e^{x + h}=e^{x}\cdot e^{h}$, the expression becomes $\frac{e^{x}\cdot e^{h}-e^{x}}{h}$.

Step3: Factor out the common term $e^{x}$

Factor out $e^{x}$ from the numerator: $\frac{e^{x}(e^{h}-1)}{h}$.

Answer:

$\frac{e^{x}(e^{h}-1)}{h}$