Question

differentiate the function.
f(x) = \frac{x}{x^{-1}+4}
which of the following shows how to find the derivative of f(x)?
a. f(x) = \frac{(x^{-1}+4)\cdot\frac{d}{dx}x - x\cdot\frac{d}{dx}(x^{-1}+4)}{(x^{-1}+4)^2}
b. f(x) = (x^{-1}+4)\cdot\frac{d}{dx}x - x\cdot\frac{d}{dx}(x^{-1}+4)
c. f(x) = \frac{x\cdot\frac{d}{dx}(x^{-1}+4)-(x^{-1}+4)\cdot\frac{d}{dx}x}{(x)^2}
d. f(x) = \frac{(x^{-1}+4)\cdot\frac{d}{dx}x - x\cdot\frac{d}{dx}(x^{-1}+4)}{x^{-1}+4}

Explanation:

Step1: Recall quotient - rule

If $f(x)=\frac{u(x)}{v(x)}$, then $f'(x)=\frac{v(x)u'(x)-u(x)v'(x)}{v(x)^2}$. Here $u(x) = x$, $v(x)=x^{-1}+4$.

Answer:

A. $\frac{(x^{-1}+4)\cdot\frac{d}{dx}x - x\cdot\frac{d}{dx}(x^{-1}+4)}{(x^{-1}+4)^2}$