Differences

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

--- chain_rules [2026/03/10 22:02] – [e.g.] hkimscil
+++ chain_rules [2026/03/10 22:11] (current) – [e.g.] hkimscil
@@ Line 45: / Line 45: @@
 \text{a} & = & \text{intercept} \\
 \text{residual} & = & (y - \widehat{y}) \\
-\dfrac{\text{d.[SSR]}^2}{\text{d.a}} & = &
+\text{SSR} & = & \sum  {\text{residual}^2} = \sum{(y - (a + b x))^2}  \\
-\dfrac{\text{d.[SSR]}^2}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.a}} \\
+\dfrac{\text{d.SSR}}{\text{d.a}} & = &
+\dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.a}} \\
 & = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.intercept}} \\
 & = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.intercept}} \\
@@ Line 60: / Line 61: @@
 \text{b} & = & \text{slope} \\
 \text{residual} & = & (y - \widehat{y}) \\
-\dfrac{\text{d.[SSR]}^2}{\text{d.a}} & = &
+\text{SSR} & = & \sum  {\text{residual}^2} = \sum{(y - (a + b x))^2}  \\
-\dfrac{\text{d.[SSR]}^2}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.b}} \\
+\dfrac{\text{d.SSR}}{\text{d.a}} & = &
+\dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.b}} \\
 & = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.b}} \\
 & = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.b}} \\
 & = & 2 * \text{residual} * -x \\
-& = & -2 * x * \text{residual} \\
+& = & -2 x * \text{residual} \\
-& = & -2 * x * (y - (a + b * x)) \\
+& = & -2 x * (y - (a + b * x)) \\
-& = & -2 * x * (y - \widehat{y}) \\
+& = & -2 x * (y - \widehat{y}) \\
 \end{eqnarray*}