Skip to content

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

\(\left\{\begin{aligned} f'' & = 0\ or\ DNE \\ f'' & change\ sign \end{aligned}\right.\)