find a function whose derivative is 20x^3 plus 6x^2 plus 5
Have you heard of an integral? Well anyway, we're looking for the anti-derivative of $\displaystyle 20x^3+6x+62+5$
In order to get an $\displaystyle x^3$ the power rule tells us we need an $\displaystyle x^4$ but the derivative of $\displaystyle x^4=4x^3$ so we'd have a 4 out in front when we want a 20, so let's put a 5 out in front of our original guess
Now the derivative of $\displaystyle 5x^4=20x^3$ so the first term is done....
Doing similar steps will lead you to an answer of $\displaystyle 20x^4+2x^3+5x+c$
Note that when we take the derivative of that, the +c has a derivative of zero, since the question is to find an antiderivative, you can choose any value for c as the answer
$\displaystyle f'(x) = 20x^3 + 6x^2 + 5$
$\displaystyle f(x) = 5x^4 + 2x^3 + 5x + C $
$\displaystyle C = konst$
$\displaystyle f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$
$\displaystyle f'(x) = a_1 + 2a_2x + 3a_3x^2 + ...+ na_nx^{n-1}$
$\displaystyle f(x) = C$ C = const. $\displaystyle \rightarrow f'(x) = 0$
$\displaystyle f(x) = kx^m \rightarrow f'(x) = m*k*x^{m-1}$