Business Calculus HW Question. If 𝑓(𝑥)=7+3/𝑥 + 2/𝑥^2 find 𝑓′ (x) Find 𝑓′(5)

Question

If 𝑓(𝑥)=7+3/𝑥 + 2/𝑥^2 find 𝑓′ (x)

Find 𝑓′(5)

Answer 1

f(x) = 7 + 3/x + 2/x²

For convenience, let's rewrite as: f(x) = 7 + 3x^-1 + 2x^-2

7 is a constant, so its derivative is 0

3x^-1 = (-1)(3x^-1-1) = -3x^-2 (By the Power Rule).

-3x^-2 = -3/x²

2x^-2 = (-2)(2x^-2-1) = -4x^-3 (By the Power Rule)

-4x^-3 = -4/x³

Then, we have that f'(x) = -3/x² - 4/x³ = -3x/x³ - 4/x³ = -(3x + 4)/x³

To find f'(5), just plug in 5 into our equation:

-(3x + 4)/x³ = -(3*5 + 4)/(5)³ = -(15 + 4)/125 = -19/125.