For a homework assignment, I was asked to find the derivative of the function $f(x)= \frac{x^{3}-6x^{2}+6}{x^{2}}.$
Using the product rule, I worked through the following: $f'(x)=\frac{\frac{d}{dx}(x^{3}-6x^{2}+6)}{x^{2}}+(x^{3}-6x^{2}+6)\left[\frac{d}{dx}\frac{1}{x^{2}}\right]$
Since the derivative of $\frac{1}{x{^2}}$ is $-\frac{2}{x^{3}}$, the left side of the formula resulted in $\frac{-2(x^{3}-6x^{2}+6)}{x^{3}}= -\frac{2x^{3}+12x^{2}-12}{x^{3}}.$
The right side of the equation split up by the product rule into $\frac{\frac{d}{dx}(x^{3})-6(\frac{d}{dx}(x^{2}))+\frac{d}{dx}(6)}{x^{2}}$
The derivatives of $x^{3}$ is $3x^{2}$; $x^{2}$ is $2x$, and 6 is 0, thus making the formula \ $\frac{3x^{2}+12x}{x^{2}}-\frac{2x^{3}-12x{^2}+12}{x^{3}}$
That's where I got stuck. Now, when I plugged the formula into WolframAlpha, the derivative calculated was $1-\frac{12}{x^{3}}$ How can I go from the formula above to the calculated result?