Suppose \(f\) is a function and I is an open interval contained in the domain of \(f\). We say that a function \(F\) is an antiderivative of \(f\) on \(I\) if \(F\) is differentiable on \(I\), and for all \(x \in I\), \(F'(x) = f(x)\)

Ref: Any difference between indefinite integral and antiderivative

The right hand derivative of a function \(f\) at a number \(a\) is the number \(f'_{+}(a)\) defined by the formula

\(f'_{+}(a) = \lim_{h \to 0^+} \dfrac{f(a+h)-f(a)}{h}\)

Similary, the left-hand derivative of \(f\) at \(a\) is

\(f'_{-}(a) = \lim_{h \to 0^-} \dfrac{f(a+h)-f(a)}{h}\)

Also other theorems, hold too:

\(f'_{+}(a) = \lim_{x \to a^+} \dfrac{f(x)-f(a)}{x-a}\)

\(f'_{-}(a) = \lim_{x \to a^-} \dfrac{f(x)-f(a)}{x-a}\)

We say that a function \(f\) is differentiable on a closed interval \([a,b]\) if \(f\) is differentiable on \((a,b)\), and the one-sided derivatives \(f'_{+}(a)\) and \(f'_{-}(b)\) are defined.

Good explanation of the need of one sided limits is given here (Apart from the similar explanation in Velleman's text)

Suppose \(f\) is a function and the closed interval \([a,b]\) is contained in the domain of \(f\). We say that a function \(F\) is an antiderivative of \(f\) on \([a,b]\) if \(F\) is differentiable on \([a,b], F'_{+}(a)=f(a), F'_{-}(b) = f(b)\), and for all \(x \in (a,b), F'(x) = f(x)\).

Suppose that \(f\) is differentiable on an interval \(I\), \(g\) is differentiable on an interval \(J\), and for every \(x \in J, g(x) \in I\). Then \(f \circ g\) is differentiable on \(J\), and for every \(x \in J\),

\((f \circ g)'(x) = f'(g(x)).g'(x)\)

where each of the derviatives in this equation is interpreted as a one-sided derivative whenever it is evaulated at an endpoint.

Suppose that \(F\) is an antiderivative of \(f\) on an interval \(I\). Then for every number \(C\), the function \(G(x) = F(x) + C\) is also an antiderivative of \(f\) on \(I\). Furthermore, these are the only antiderivatives. In other words, if \(G\) is any antiderivative of \(f\) on \(I\), then there is some number \(C\) such that for all \(x \in I\), \(G(x) = F(x) + C\).

Suppose \(r\) is a rational number and \(r \neq -1\), and let \(f(x) = x^r\). Then the function

\(F(x) = \dfrac{x^{r+1}}{r+1}\) is an antiderivative of \(f\) on every interval contained in the domain of \(f\).

To list these intervals explicitly, we first write the rational number \(r\) in lowest terms as \(r = \dfrac{m}{n}\).

If \(n\) is odd and \(r \geq 0\), then \(f(x) = x^r\) is defined for all \(x\), and \(F\) is an antiderivative of \(f\) on the interval \((-\infty, \infty)\).

If \(n\) is odd and \(r < 0\), then \(f(x)\) is defined for all values of \(x\) except \(x = 0\), and \(F\) is an antiderivative of \(f\) on each of the intervals \((-\infty, 0)\) and \((0, \infty)\).

If \(n\) is even, then the domain of \(f\) is \([0, \infty)\) if \(r > 0\), and \((0, \infty)\) if \(r < 0\). In both of these cases, \(F\) is an antiderivative of \(f\) on the entire domain of \(f\).

Suppose that \(F\) and \(G\) are antiderivatives of \(f\) and \(g\), respectively, on an interval \(I\). Then:

• \(F + G\) is an antiderivative of \(f + g\) on \(I\).
• For any number \(c\), \(cF\) is an antiderivative of \(cf\) on \(I\).
• \(F - G\) is an antiderivative of \(f-g\) on \(I\).