Chain rule
#[If $f$ is a differentiable function of $u$ and $u$ is a differentiable function of $x,$ then the composite $f(u)$ is a differentiable function of $x,$ and][Si $f$ es una función dierenciable de $u$ y $u$ es una función dierenciable de $x,$ entonces el compuesto $f(u)$ es una función diferenciable de $x,$ y]#
$\dfrac{d}{dx}[f(u)] = f'(u) \dfrac{du}{dx} \qquad$ #[Chain rule][Regla de la cadena]#
Chain rule in words:
#[Examples][Ejemplos]#
1. $\dfrac{d}{dx}(u^{3}) = 3u^2\dfrac{du}{dx}. \qquad$
$\dfrac{d}{dx}[\color{indianred}{(4x-8)}^{3}] = 3(4x-8)^2\dfrac{d}{dx}\color{indianred}{(4x-8)}$
$\qquad = 3(4x-8)^2(4) = 12(4x-8)^2 $
2. $\dfrac{d}{dx}\sqrt{u} = \dfrac{1}{2\sqrt{u}}\dfrac{du}{dx} \qquad$
$\dfrac{d}{dx}\sqrt{\color{indianred}{x^2-1}}= \dfrac{1}{2\sqrt{\color{indianred}{x^2-1}}}\dfrac{d}{dx}\color{indianred}{(x^2-1)}$
$\qquad = \dfrac{1}{\sqrt{x^2-1}}(2x) = \dfrac{2x}{\sqrt{x^2-1}}$
3. $\dfrac{d}{dx}|u| = \dfrac{|u|}{u}\dfrac{du}{dx} \qquad$
$\dfrac{d}{dx}|\color{indianred}{x^3-1}|= \dfrac{|\color{indianred}{x^3-1}|}{\color{indianred}{x^3-1}}\dfrac{d}{dx}\color{indianred}{(\color{indianred}{x^3-1})}$
$\qquad = \dfrac{3x^2|x^3-1|}{x^3-1}$
#[Some for you][Algunos para ti]#
Calculate the given derivative. Unless otherwise stated, you need not simplify the answer completely, but certain obvious simplifications are required (multiplication by 1, multiplication by 0, double minus sign, etc.).
$\displaystyle y = %1$
$\displaystyle \frac{dy}{d%0}=$ BOX*
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$\displaystyle y = %3$
$\displaystyle \frac{dy}{d%2}=$ BOX*
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$\displaystyle y = %5$
$\displaystyle \frac{dy}{d%4}=$ BOX*
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$\displaystyle y = %7$
$\displaystyle \frac{dy}{d%6}=$ BOX*
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