Differentiation
Derivative¶
\(f'(x_0) = \lim\limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x} = \lim\limits_{\Delta x \to 0} \dfrac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}\)
\(f'_-(x_0) = \lim\limits_{\Delta x \to 0^-} \dfrac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \lim\limits_{x \to x_0^-} \dfrac{f(x) - f(x_0)}{x - x_0}\)
\(f'_+(x_0) = \lim\limits_{\Delta x \to 0^+} \dfrac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \lim\limits_{x \to x_0^+} \dfrac{f(x) - f(x_0)}{x - x_0}\)
\(f'(x_0)\ exists \iff f'_-(x) = f'_+(x)\)
Algorithms¶
\([f(x) \pm g(x)]' = f'(x) \pm g'(x)\)
\([f(x) \cdot g(x)]' = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)
\([\dfrac{f(x)}{g(x)}]' = \dfrac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\)
Differentiation Rules¶
\((C)' = 0\)
\(\boxed{(x^n)' = nx^{n-1}}\)
\(\boxed{(\sin x)' = \cos x}\)
\(\boxed{(\cos x)' = -\sin x}\)
\(\boxed{(\tan x)' = \sec^2 x}\)
\(\boxed{(\sec x)' = \sec x \cdot \tan x}\)
\((\cot x)' = - \csc^2 x\)
\((\csc x)' = - \csc x \cdot \cot x\)
\(\boxed{(a^x)' = a^x \cdot \ln a}\)
\((e^x)' = e^x\)
\(\boxed{(\log_ax)' = \frac{1}{x \cdot \ln a}}\)
\((\ln x)' = \frac{1}{x}\)
The Chain Rule¶
\(\boxed{(f(g(x)))' = f'(g(x)) \cdot g'(x)}\)
Inverse Functions¶
Inverse Functions
\(f^{-1}(f(x)) = x \iff f(f^{-1}(x)) = x\)
if \(\ g(x) = f^{-1}(x)\),
then \(\ \boxed{g'(x) = \dfrac{1}{f'(g(x))}},\ f'(g(x)) \neq 0\)
L'Hôpital's Rule¶
when \(x \to a\),
if \(\left\{\begin{aligned} f(x) & \to 0 \\ g(x) & \to 0 \end{aligned}\right.\ or \left\{\begin{aligned} f(x) & \to \infty\\ g(x) & \to \infty \end{aligned}\right.\),
then \(\boxed{\lim\limits_{x \to a} \dfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \dfrac{f'(x)}{g'(x)}}\)
Intermediate Value Theorem¶
If \(f\) is continuous on \([a, b]\), \(u\) is a number such that \(\min(f(a), f(b)) < u < \max(f(a), f(b))\),
then there is a \(c \in (a, b)\) such that \(f(c) = u\)
Mean Value Theorem¶
If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then
\(\exists c \in (a, b)\) such that \(\boxed{f'(c) = \dfrac{f(b)-f(a)}{b-a}}\)
Extreme Value Theorem¶
If \(f\) is continuous on the \([a, b]\), then \(f\) must attain a maximum and a minimum, each at least once, i.e.
\(\exists c, d \in [a, b]\) such that \(\boxed{f(c) \leq f(x) \leq f(d), \forall x \in [a, b]}\)
Concavity¶
- If \(\boxed{f'' > 0}\), then \(f\) concave up
- If \(\boxed{f'' < 0}\), then \(f\) concave down
Points of Inflection
\(\left\{\begin{aligned} f'' & = 0\ or\ DNE \\ f'' & change\ sign \end{aligned}\right.\)