Differentiation

Differentiation#

Description	Equations
Derivative as a function	\(f'(x) = \lim\limits_{h \to 0}\dfrac{f(x+h) - f(x)}{h}\)
Geometric interpretation of derivatives	The tangent line of \(y = f(x)\) at \((a, f(a))\) has a slope of \(f'(a)\) \(m = \lim\limits_{x \to a} \dfrac{f(x) - f(a)}{x-a} = f'(a)\)
Derivatives and instantaneous rate of change	The derivative \(f'(a)\) is the instantaneous rate of change of \(y=f(x)\) with respect to \(x\) when \(x=a\): \(v(a) = \lim\limits_{h \to 0} \dfrac{f(a+h) - f(a)}{h} = f'(a)\)
Differentiation and continuity	If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).
Non-differentiable conditions	1. a corner 2. a discontinuity 3. a vertical tangent

Description	Equations
Constant multiple rule	\(\frac{d}{dx}[cf(x)] = c\frac{d}{dx}f(x)\)
Addition and subtraction rule	\(\frac{d}{dx}[f(x)\pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)\)
Product rule	\(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
Quotient rule (best practice: use product rule)	\(\frac{d}{dx}\dfrac{f(x)}{g(x)} = \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\)
Constant rule	\(\frac{d}{dx}c = 0\)
Power rule	\(\frac{d}{dx} x^n = nx^{n-1}\)
Chain rule	\(\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \newline \frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x)\)
Linear approximations	\(f(x) \approx f(a) + f'(a)(x-a)\)
Differentials	\(dy = f'(x) \ dx\)

Description	Equations
Limit associated with sine	\(\lim\limits_{\theta \to 0}\dfrac{\sin\theta}{\theta} = 1\)
Limit associated with cosine	\(\lim\limits_{\theta\to 0}\dfrac{\cos\theta - 1}{\theta} = 0\)
Definition of \(e\)	\(\lim\limits_{h\to 0}\dfrac{e^h - 1}{h} = 1\)
\(e\) as a limit	\(e = \lim\limits_{x\to 0} (1+x)^{1/x}\)
\(e\) as a limit	\(e = \lim\limits_{n\to \infin} (1+\frac{1}{n})^{n}\)

Function \(f(x)\)	Derivative \(f'(x)\)	Function \(f(x)\)	Derivative \(f'(x)\)
\(c\)	\(0\)	\(x^n\)	\(nx^{n-1}\)
\(x\)	\(1\)	\(\lvert x \rvert\)	\(\dfrac{x}{\lvert x \rvert}\)
\(e^x\)	\(e^x\)	\(\ln x\)	\(\dfrac{1}{x}\)
\(a^x\)	\(a^x\ln(a)\)	\(\log_{a}x\)	\(\dfrac{1}{x\ln(a)}\)
\(\sin x\)	\(\cos x\)	\(\sec x\)	\(\sec x \tan x\)
\(\cos x\)	\(-\sin x\)	\(\csc x\)	\(-\csc x \cot x\)
\(\tan x\)	\(\sec^2 x\)	\(\cot x\)	\(-\csc^2 x\)
\(\arcsin x\)	\(\dfrac{1}{\sqrt{1-x^2}}\)	\(\arctan x\)	\(\dfrac{1}{1+x^2}\)
\(\arccos x\)	\(\dfrac{-1}{\sqrt{1-x^2}}\)