Question

find f(x). f(x)=\frac{x}{x - 34} f(x)=square (type an exact answer.)

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $f(x)=\frac{u(x)}{v(x)}$, then $f^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^2}$. Here, $u(x) = x$, so $u^{\prime}(x)=1$, and $v(x)=x - 34$, so $v^{\prime}(x)=1$.

Step2: Substitute into the quotient - rule formula

$f^{\prime}(x)=\frac{1\times(x - 34)-x\times1}{(x - 34)^2}$.

Step3: Simplify the numerator

Expand the numerator: $1\times(x - 34)-x\times1=x - 34-x=-34$.
So, $f^{\prime}(x)=\frac{-34}{(x - 34)^2}$.

Answer:

$\frac{-34}{(x - 34)^2}$