Can someone explain to me how to approach this problem and what the objective of a problem like this is? Calculate the derivative. 
$$\frac {d}{dx} [ 12 x^{5/3} - 7 x^{-2} -12 ]$$
I have no prior knowledge of calculus and am struggling terribly to understand this conceptually. I really have no idea how to even begin this problem. 
 A: The idea is to apply the basic rules of differentiation:
$$\begin{align}
&\frac {d}{dx} [ 12 x^{5/3} - 7 x^{-2} -12 ]\\
=&\;\frac {d}{dx} [ 12 x^{5/3}] - \frac {d}{dx} [7 x^{-2}] -\frac {d}{dx} [12 ]\tag{sum rule}\\
=\;&\frac {d}{dx} [12x^{5/3}] - \frac {d}{dx} [7 x^{-2}] -0\tag{constant rule}\\
=\;&12\frac {d}{dx} [x^{5/3}] - 7\frac {d}{dx} [x^{-2}] \tag{constant multiple rule}\\
=\;&12\frac{5}{3}(x^{2/3}) - 7(-2)(x^{-3}) \tag{power rule}\\
=\;&20x^{2/3} +14x^{-3} \tag{simplification}\\
\end{align}$$
This site can be useful for you.
A: Some rules for differentiation ($D$ is the differential operator):
$$\begin{align}D(f(x) + g(x)) &= Df(x) + Dg(x) & \text{($D$ is additive)}\\ D(af(x)) &= aDf(x) & \text{($D$ is degree $1$ homogeneous)} \\ D(1) &= 0 \\ D(x^k) &= kx^{k-1}\end{align}$$
Now let's use these on your problem:
$$\begin{align}D(12x^{5/3}−7x^{−2}−12) &= D(12x^{5/3}) + D(-7x^{-2}) + D(-12) \\ &= 12D(x^{5/3})-7D(x^{-2})-12D(1) \\ &=12\left(\frac 53\right)x^{2/3} -7(-2x^{-3})-12(0) \\ &=20x^{2/3}+14x^{-3}\end{align}$$
A: Some basics:
$$\frac{\mathrm d}{\mathrm dx}[f(x) + g(x)] = \frac{\mathrm d}{\mathrm dx}f(x) + \frac{\mathrm d}{\mathrm dx}g(x) $$
$$\frac{\mathrm d}{\mathrm dx}c = 0\quad\hbox{(c is a constant)}$$
$$\frac{\mathrm d}{\mathrm dx}c\times f(x) = c \frac{\mathrm d}{\mathrm dx}f(x)$$
$$\frac{\mathrm d}{\mathrm dx}x^p = px^{(p-1)}$$
$$\frac{\mathrm d}{\mathrm dx}f(x)g(x) = \left[\frac{\mathrm d}{\mathrm dx}f(x)\right]\times g(x) + \left[\frac{\mathrm d}{\mathrm dx}g(x)\right]\times f(x)$$
I know that last one is hard to remember but you won't need it for this problem.
Just apply the appropriate rules and it is straightforward.
