Question

find the derivative of the function. f(z)=e^z/(z - 5) f(z)= find the derivative of the function. f(t)=e^4t*sin(2t) f(t)=

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. For $f(z)=\frac{e^{z}}{z - 5}$, let $u = e^{z}$ and $v=z - 5$.

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

The derivative of $u = e^{z}$ with respect to $z$ is $u'=e^{z}$, and the derivative of $v=z - 5$ with respect to $z$ is $v' = 1$.

Step3: Apply quotient - rule

$f'(z)=\frac{e^{z}(z - 5)-e^{z}\times1}{(z - 5)^{2}}=\frac{e^{z}(z - 5-1)}{(z - 5)^{2}}=\frac{e^{z}(z - 6)}{(z - 5)^{2}}$

for second function:

Step1: Recall product - rule

The product - rule states that if $y = uv$, then $y'=u'v+uv'$. For $F(t)=e^{4t}\sin(2t)$, let $u = e^{4t}$ and $v=\sin(2t)$.

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

Using the chain - rule, the derivative of $u = e^{4t}$ with respect to $t$ is $u'=4e^{4t}$, and the derivative of $v=\sin(2t)$ with respect to $t$ is $v' = 2\cos(2t)$.

Step3: Apply product - rule

$F'(t)=4e^{4t}\sin(2t)+2e^{4t}\cos(2t)=2e^{4t}(2\sin(2t)+\cos(2t))$

Answer:

$\frac{e^{z}(z - 6)}{(z - 5)^{2}}$