Differentiation#

Derivatives#

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

Differentiation rules#

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

\(dy = f'(x) \ dx\)

Special limits#

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

\(e = \lim\limits_{n\to \infin} (1+\frac{1}{n})^{n}\)

Table of derivatives#

Function \(f(x)\)

Derivative \(f'(x)\)

Function \(f(x)\)

Derivative \(f'(x)\)

\(c\)

\(0\)

\(x^n\)

\(nx^{n-1}\)

\(e^x\)

\(e^x\)

\(\ln x\)

\(\dfrac{1}{x}\)

\(\sin x\)

\(\cos x\)

\(\cos x\)

\(-\sin x\)

\(\tan x\)

\(\sec^2 x\)

\(\arcsin x\)

\(\dfrac{1}{\sqrt{1-x^2}}\)

\(\arctan x\)

\(\dfrac{1}{1+x^2}\)