Differences

This shows you the differences between two versions of the page.

--- chain_rules [2025/08/22 12:34] – [E.g] hkimscil
+++ chain_rules [2025/08/22 13:19] (current) – [e.g.] hkimscil
@@ Line 16: / Line 16: @@
 y & = & t^2 \\
 t & = & 2x^2 + 1 \\
-\frac{dy}{dt} & = & 2t \\
+\\
-& = & 2 (2x^2 + 1) \\
+&\phantom{=}\, \frac{dy}{dt} & = 2t \\
-& = & (4x^2 + 2) \\
+&\phantom{=}\, & = 2 (2x^2 + 1) \\
-\frac{dt}{dx} & = & 4x \\
+&\phantom{=}\, & = (4x^2 + 2) \\
-\therefore{}
+\\
+&\phantom{=}\, \frac{dt}{dx} & = 4x \\
+\\
 \frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\
 & = & (4x^2 + 2) * 4x \\
@@ Line 28: / Line 30: @@
 ====== 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