Derivadas

Definição

(1)
\frac{d}{dx} f(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}

Regras

Derivada de uma constante

(2)
\begin{align} \frac{dc}{dx} &= \lim_{h\to0} \frac{c - c}{h} \ &= \lim_{h\to0} \frac{0}{h} \ &= \lim_{h\to0} 0 \ &= 0 \end{align}

Derivada da função identidade

(3)
\begin{align} \frac{dx}{dx} &= \lim_{h\to0} \frac{x+h - x}{h} \ &= \lim_{h\to0} \frac{h}{h} \ &= \lim_{h\to0} 1 \ &= 1 \end{align}

Trigonométricas

Se algumas passagens parecerem obscuras, revise algumas propriedades trigonométricas.

Seno

Dado1:

(4)
\lim_{h\to0} \frac{\sin h}{h} = 1

Temos que:

(5)
\begin{align} \frac{d}{dx} \sin x &= \lim_{h\to0} \frac{\sin (x+h) - \sin x}{h} \ &= \lim_{h\to0} \frac{\sin x \cdot \cos h + \sin h \cdot \cos x - sin x}{h} \ &= \lim_{h\to0} \frac{\sin x \cdot \cos h - \sin x}{h} + \lim_{h\to0} \frac{\sin h \cdot \cos x}{h} \ &= \sin x \cdot \lim_{h\to0} \frac{\cos h - 1}{h} + \cos x \cdot \lim_{h\to0} \frac{\sin h}{h} \ &= \sin x \cdot \lim_{h\to0} \frac{\cos h - 1}{h} \cdot \frac{\cos h + 1}{\cos h + 1} + \cos x \cdot 1 \ &= \sin x \cdot \lim_{h\to0} \frac{\cos^2 h - 1}{h (\cos h + 1)} + \cos x \ &= \sin x \cdot \lim_{h\to0} \frac{-\sin^2 h}{h (\cos h +1)} + \cos x \ &= \sin x \cdot \lim_{h\to0} -\sin h \cdot \lim_{h\to0} \frac{\sin h}{h} \cdot \lim_{h\to0} \frac{1}{\cos h + 1} + \cos x \ &= \sin x \cdot 0 \cdot 1 \cdot \frac{1}{2} + \cos x \ &= \cos x \end{align}

Cosseno

Sabendo que:

(6)
\begin{align} \cos x &= \sin \left(x + \frac{\pi}{2}\right)\ \frac{d}{dx} \cos x &= \frac{d}{dx} \sin \left(x + \frac{\pi}{2}\right) \end{align}

Sendo u = x + \frac{\pi}{2} e usando a regra da composição:

(7)
\begin{align} \frac{d}{dx} \cos x = \frac{d}{dx} \sin (u) &= \frac{d}{du}\sin u\cdot \frac{du}{dx} \ &= \frac{d}{du}\sin u\cdot \frac{d}{dx} \left(x + \frac{\pi}{2}\right) \ &= \cos u \cdot 1 \ &= \cos \left(x + \frac{\pi}{2}\right) \ &= -\sin x \end{align}
Footnotes
Page tags: cálculo
page_revision: 2, last_edited: 1238339108|%e %b %Y, %H:%M %Z (%O ago)
EditTags History Files Print Site tools+ Options
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License