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Definition: Differentiability & Derivative
Let $f $ be a real-valued function defined on an open interval containing a point $a $. We say $f $ is differentiable at $a $, or $f $ has a derivative at $a$, or $f $ has a derivative at $a $ if the limit
$$\limu{ x }{ a } \frac{ f(x) - f(a)}{ x - a } $$
exists and is finite. This value is $f’(a) $ i.e. the derivative of $f$ at $a$.
Theorem
28.228.2
$f$ is differentiable at $a \implies f$ is continuous at $a$.
Theorem
28.328.3
$f $ and $g$ are differentiable at $a \implies cf, f+ g, fg, f/g \pare{g(a) \neq 0}$ are all differentiable at $a $
- $(cf)^\prime(a) = c \cdot f’(a)$
- $(f + g) ^ \prime (a) = f’(a) + g’(a)$
- $(fg)^ \prime(a) = f(a) g’(a) + f’(a) g(a)$
Theorem
28.4: Chain Rule28.4
$f $ is differentiable at $a $ and $g $ is differentiable at $f(a) \implies g \circ f $ is differentiable at $a $
$(g \circ f)^ \prime (a) = g’(f(a)) \cdot f’(a)$.