find the derivative of the function. f(z) = e^z/(z - 7) f(z) =

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = e^{z}$ and $v=z - 7$.

Step2: Find $u'$ and $v'$

The derivative of $u = e^{z}$ with respect to $z$ is $u'=e^{z}$, and the derivative of $v=z - 7$ with respect to $z$ is $v' = 1$.

Step3: Substitute into quotient - rule

$f'(z)=\frac{e^{z}(z - 7)-e^{z}\times1}{(z - 7)^{2}}=\frac{e^{z}(z - 7-1)}{(z - 7)^{2}}=\frac{e^{z}(z - 8)}{(z - 7)^{2}}$

Answer:

$\frac{e^{z}(z - 8)}{(z - 7)^{2}}$