التفاضل والتكامل Formulas

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}$

$\frac{d}{dx}(x^n) = n x^{n-1}$

$(fg)' = f'g + fg'$

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

$\frac{d}{dx}\sin x = \cos x$

$\frac{d}{dx}\cos x = -\sin x$

$\frac{d}{dx} e^x = e^x$

$\frac{d}{dx} \ln x = \frac{1}{x}$

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\quad(n \neq -1)$

$\int \frac{1}{x}\,dx = \ln|x| + C$

$\int \sin x\,dx = -\cos x + C,\quad \int \cos x\,dx = \sin x + C$

$\int e^x\,dx = e^x + C$

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$