Differences

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--- chain_rules [2025/08/04 21:10] – created hkimscil
+++ chain_rules [2025/08/22 13:19] (current) – [e.g.] hkimscil
@@ Line 5: / Line 5: @@
 y & =& f(g(x)) \\
 \frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx}  \\
-&  & \frac {dy}{dt} = f'(t) = f'(g(x)) \\
+&  & \frac {dy}{dt} = f'(t) = f'(g(x)) \;\; \text{and  } \\
-&  & \because{ \frac {dt}{dx} = g'(x) } \\
+&  & \frac {dt}{dx} = g'(x) \\
-&  & \frac {dy}{dx} = f'(g(x)) * g'(x) \\
+\therefore{ \;\; } \frac {dy}{dx} & = & f'(g(x)) * g'(x) \\
 \end{eqnarray*}
+====== E.g ======
 \begin{eqnarray*}
 y & = & (2x^2 + 1)^2 \\
@@ Line 15: / Line 16: @@
 y & = & t^2 \\
 t & = & 2x^2 + 1 \\
-y' & = & 2t \\
+\\
-t' & = & 4x \\
+&\phantom{=}\, \frac{dy}{dt} & = 2t \\
-\frac{dy}{dx} & = & \frac{dy}{dt} * \frac{dt}{dx} \\
+&\phantom{=}\, & = 2 (2x^2 + 1) \\
-\frac{dy}{dx} & = & 2(t^{2-1}} * 4 x^{2-1} \\
+&\phantom{=}\, & = (4x^2 + 2) \\
-& = & 2(2x^2 + 1) * 4x \\
+\\
+&\phantom{=}\, \frac{dt}{dx} & = 4x \\
+\\
+\frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\
 & = & (4x^2 + 2) * 4x \\
-& = & 16x^3 + 8x \\ -
+& = & 16x^3 + 8x \\
-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*}
+====== e.g. ======
+see [[:gradient descent]]
+\begin{eqnarray*}
+\because{ \;\; } \text{predicted value } \; \hat{y} & = & a + b x \\
+\text{and }\;\;  \text{residual} & = & y - \hat{y} \\
+\therefore{} \;\; \text{residual}^2 & = & (y - (a + b x)) \\
+\therefore{} \sum{\text{residual}^2} & = & \sum{(y - (a + b x))^2} \\
+& = & \text{SSE,  sum of square residuals} \\
+\\
+\dfrac{\text{dSSE}}{\text{da}} & = &  \\
+\end{eqnarray*}
+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