What is the derivative of the function f(x) = (3x - 8)/e^x?

To find the derivative of the function ( f(x) = \frac{3x - 8}{e^x} ), we can apply the quotient rule, which states that if you have a function ( f(x) = \frac{g(x)}{h(x)} ), then its derivative is given by:

[

f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}

]

In this case, ( g(x) = 3x - 8 ) and ( h(x) = e^x ). Let's calculate the derivatives of ( g(x) ) and ( h(x) ):

• The derivative of ( g(x) ):

[

g'(x) = 3

]

• The derivative of ( h(x) ):

[

h'(x) = e^x

]

Now we plug these derivatives into the quotient rule formula:

[

f'(x) = \frac{(3)(e^x) - (3x - 8)(e^x)}{(e^x)^2}

]

We can simplify the expression in the numerator