Chain rule (Leibniz notation)
If $y$ is a differentiable function of $u$ and, in turn, $u$ is a differentiable function of $x$, then $y$ is a differentiable function of $x$, and:
$\displaystyle \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$ \gap[30] \t
#[Examples][Ejemplos]#
1. #[If][Si]# $y = u^3$ #[and][y]# $u = 2x + 3x^4$, #[then][entonces]#
$\displaystyle \frac{dy}{dx}$ \t $\displaystyle {}= \frac{dy}{du} \frac{du}{dx}$ \gap[30] \t
\\ \t $\displaystyle {}= 3u^2(2+12x^3)$
\\ \t $\displaystyle {}= 3(2x + 3x^4)^2(2+12x^3)$ \gap[30] \t
2. #[If][Si]# $A = \pi r^2$ #[and][y]# $r = 3t+1$, #[then][entonces]#
$\displaystyle \frac{dA}{dt}$ \t $\displaystyle {}= \frac{dA}{dr} \frac{dr}{dt}$ \gap[30] \t
\\ \t $\displaystyle {}= 2\pi r(3) = 6\pi r$
\\ \t $\displaystyle {}= 6\pi (3t+1)$ \gap[30] \t