Suppose that we have two functions, f(x) and g (x), and that
Calculate the values of the following derivatives when x is equal to 3.
- f (3) = 4,
- g (3) = 5,
- f '(3) = -4, and
- g '(3) = 5.
Hello, Merdiemae!
We need the Quotient Rule . . .Suppose that we have a function, $\displaystyle f(x)$, and that: .$\displaystyle f(3) = 4,\;\;f'(3) = \text{-}4$
Calculate the value of $\displaystyle \frac{d}{dx}\left(\frac{f(x)}{x}\right)$ when $\displaystyle x = 3$
. . $\displaystyle \frac{d}{dx}\left(\frac{f(x)}{x}\right) \;=\;\frac{x\!\cdot\!f'(x) - 1\!\cdot\!f(x)}{x^2}$
Then: .$\displaystyle \frac{d}{dx}\left(\frac{f(x)}{x}\right)\bigg]_{x=3} \;=\; \frac{3\!\cdot\!f'(3) - 1\!\cdot\!f(3)}{3^2} \;=\;\frac{3(\text{-}4) - 1(4)}{3^2} \;=\;\boxed{-\frac{16}{9}}$