[Lecture 12] Differentiability Theorem

Let \(f\) and \(g\) be real functions and let \(\alpha \in \mathbb{R}\). If \(f\) and \(g\) are differentiable at \(a\). then

1. (Linearity) \(f \pm g)'(a) = f'(a) + g'(a)\).
2. (Product Rule) \(fg)'(a) = f'(a)g(a) + f(a)g'(a)\).
3. (Quotient Rule) \(\frac{f}{g})'(a) = \frac{f'(a)g(a) + f(a)g'(a)}{(g(a))^2}\).
4. (Chain Rule) \(f \circ g)'(a) = f'(g(a))g'(a)\).