y = uv \frac{d y}{d x} = v \frac{d u}{d x} + u \frac{d v}{d x}

Examples

$y = x \sin x$

y = x sin x u = x v = sin x \frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x} \frac{d y}{d x} = x (cos x) + sin x \frac{d y}{d x} = x cos x + sin x

y = x^{4} e^{3 x} u = x^{4} v = e^{3 x} \dots

Todo
Complete.

y = ln x cos x u = ln x v = cos x \frac{d u}{d x} = \frac{1}{x} \frac{d v}{d x} = - sin x \frac{d y}{d x} = cos x \times \frac{1}{x} + ln x - sin x \frac{d y}{d x} = \frac{c o s x}{x} - sin x ln x