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chain_rules [2025/08/04 21:10] – created hkimscilchain_rules [2025/08/05 00:01] (current) – [e.g.] hkimscil
Line 13: Line 13:
 y & = & (2x^2 + 1)^2 \\ y & = & (2x^2 + 1)^2 \\
 t & = & 2x^2 + 1 \\ t & = & 2x^2 + 1 \\
-y & = & t^2 \\+y & = & t^2 \\b
 t & = & 2x^2 + 1 \\ t & = & 2x^2 + 1 \\
-y' & = & 2t \\ +\frac{dy}{dt} & = & 2t \\ 
-t' & = & 4x \\ +& = & 2 (2x^2 + 1) \\ 
-\frac{dy}{dx} & = & \frac{dy}{dt} * \frac{dt}{dx} \\ +& = & (4x^2 + 2) \\  
-\frac{dy}{dx} & = & 2(t^{2-1}} * 4 x^{2-1\\ +\frac{dt}{dx} & = & 4x \\ 
-& = & 2(2x^2 + 1) * 4x \\+\therefore{} 
 +\frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\ 
 & = & (4x^2 + 2) * 4x \\ & = & (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*} \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
  
chain_rules.1754309400.txt.gz · Last modified: 2025/08/04 21:10 by hkimscil

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