====== Chain rules ======
\begin{eqnarray*}
y & = & f(t) \\
t & = & g(x) \\
y & =& f(g(x)) \\
\frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx}  \\
&  & \frac {dy}{dt} = f'(t) = f'(g(x)) \\
&  & \because{ \frac {dt}{dx} = g'(x) } \\
&  & \frac {dy}{dx} = f'(g(x)) * g'(x) \\
\end{eqnarray*}

\begin{eqnarray*}
y & = & (2x^2 + 1)^2 \\
t & = & 2x^2 + 1 \\
y & = & t^2 \\b
t & = & 2x^2 + 1 \\
\frac{dy}{dt} & = & 2t \\
& = & 2 (2x^2 + 1) \\
& = & (4x^2 + 2) \\ 
\frac{dt}{dx} & = & 4x \\
\therefore{}
\frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\ 
& = & (4x^2 + 2) * 4x \\
& = & 16x^3 + 8x \\
\end{eqnarray*}

====== e.g. ======
see [[:gradient descent]]

y.hat = a + b * x 
a = intercept 
residuals = (y - y.hat)
d.sum.of.residuals^2 / d.intercept 
= d.sum.of.residuals^2 / d.sum.of.residuals * d.sum.of.residuals / d.intercept
= (2 * residual) *  d(y - y.hat)/d.intercept
= (2 * residual) *  d(y - (a + bx))
= (2 * residual) *  d(y - a - bx)
= (2 * residual) *  -1
= -2 * residual

y.hat = a + b * x 
b = slope
d.sum.of.square.res / d.slope
= d.sum.of.square.res / d.sum.of.res * d.sum.of.res / d.slope
= d.sum.of.square.res / d.slope
= (2 * residual) * d(y - a - bx)
= (2 * residual) * - x
= - 2 * x * residual