chain_rules
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chain_rules [2025/08/04 21:10] – created hkimscil | chain_rules [2025/08/05 00:01] (current) – [e.g.] hkimscil | ||
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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' | + | \frac{dy}{dt} |
- | t' | + | & = & 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' | + | |
- | & & \because{ \frac {dt}{dx} = g'(x) } \\ | + | |
- | & & \frac {dy}{dx} = f' | + | |
\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)/ | ||
+ | = (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