Chapter review “ Find the indicated derivative”

Question

Find $y'$ if $$y = 6x^2 - 10 + \frac{12}{x^2}$$ I feel like this is an easy question but I am drawing a blank on how to do it. If someone could help explain in steps how to do this problem so I can do the next similar one would be greatly appreciated.

Answer 1

We can apply the simple properties of differentiation as such: $$y = 6x^2 - 10 + \frac{12}{x^2}$$ $$ y'= 6(2x)-0+12 \cdot \frac{-2}{x^3}$$ I leave it to you to figure out the "simple properties"...

Answer 2

Apply the power rule for derivatives: $$\frac{d}{dx}(x^r)=rx^{r-1},$$

and the sum/subtraction rules for derivatives:

$$\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x)),$$ $$\frac{d}{dx}(f(x)-g(x))=\frac{d}{dx}(f(x))-\frac{d}{dx}(g(x)).$$

Where

$$r=2: \quad \frac{d}{dx}(6x^2)=6\cdot \frac{d}{dx}(x^2)=6( 2x)=12x,$$ $$r=-2: \quad \frac{d}{dx}(12x^{-2})=12\cdot\frac{d}{dx}(x^{-2})=12(-2x^{-3})=-24x^{-3},$$

and the derivative of any constant function is zero. So, $\frac{d}{dx}(10)=0$. Can you finish?