Q:

How to solve complex derivatives with multiple terms?

I have several derivatives to find:

For $g(s)=3s^3-s+4$, $g'(s)=$
For $p(t)=\frac1t+t^2$, $\frac{\mathrm dp}{\mathrm dt}=$
For $w(u)=\sqrt u-2u^2-10$, $\frac{\mathrm dw}{\mathrm du}=$

Each has multiple terms, and as a result is difficult to determine how to solve.
I need to find out how to go about solving these, for example I can't use the power rule because these functions have more than just one term.
I am looking for insight on what steps to take.

A:

Hints:
All three problems can be solved by combining three facts:

$D_x(f(x)\pm g(x))=D_x(f(x))\pm D_x(g(x))$
$D_x(cf(x))=cD_x(f(x))$ for any constant $c$
$D_x(x^n)=nx^{n-1}$ for any $n\neq 0$ (including $n=-1$ and $n=\frac12$.)

In the first problem, the submitted solution takes −s and replaces it with +s, which certainly isn't the derivative of $-s$ with respect to $s$. In the second solution, it looks like the user doesn't know what the derivative of 1/t is, and in the third solution looks overcomplicated and contains an x for some reason.

