1.9 Composite Functions and Chain Rule Differentiation
Let \(z = f(x,y)\) where \(x = g(r,s)\) and \(y = (r,s)\) , so that \(z\) is a function of \(r\) and \(s\). Then \[\frac {\partial z}{\partial r} =\frac {\partial z}{\partial x}\cdot \frac {\partial x}{\partial r} + \frac {\partial z}{\partial y}\cdot \frac {\partial y}{\partial r}\]
In general if \(u = F(x_1,x_2,\cdots \cdots \cdots , x_n)\) where \(x_1 = f_1 (r_1,r_2,\cdots \cdots \cdots , r_n)\) then \[\frac {\partial u}{\partial r_1}=\frac {\partial u}{\partial x_1}\frac {\partial x_1}{\partial r_1}+ \frac {\partial u}{\partial x_2} \cdot \frac {\partial x_2}{\partial r_1}+\cdots \cdots \cdots + \frac {\partial u}{\partial x_n}\cdot \frac {\partial x_n}{\partial r_1}\]
In particular if \(x_1,x_2,\cdots \cdots \cdots , x_n\) depend only on one variable ,\(s\), then \[\frac {du}{ds} = \frac {\partial u}{\partial x_1}\cdot \frac {dx_1}{ds}+ \frac {\partial u}{\partial x_2}\cdot \frac {dx_2}{ds} +\cdots \cdots \cdots +\frac {\partial u}{\partial x_n}\cdot \frac {dx_n}{ds}\]
These results are called chain rule and are useful in transforming derivatives from one set of variables to another.
If \(f(x) = \sin x\hspace {0.4cm}, \hspace {0.4cm} x = t^2\hspace {1cm} -\infty < t < \infty \)
Then \(\hspace {0.4cm} f(x(t)) = \sin (t^2) = F(t)\) \begin {align*} \frac {dF}{d t} & = \frac {d F}{dx}\cdot \frac {d x}{dt}\\ & = \cos x \cdot 2t\\ & = 2t\cos (t^2)\\ \end {align*}
Let \(r(x,y) =x^2y - e^{2y}\)
where\(\hspace {0.4cm} x = 3t^2\hspace {1cm} y = \sin t \hspace {1cm} 1< t <4\)
Then \(\hspace {0.5cm} r\big (x(t),y(t)\big ) = \hspace {0.5cm} = R(t)\)
\begin {align*} \frac {dR}{dt} & = \frac {\partial R}{\partial x}\cdot \frac {dx}{dt} + \frac {\partial R}{\partial y}\cdot \frac {dy}{dt}\\ & = 2xy\cdot 6t + \big (x^2 - 2e^{2y}\big )\cdot \cos t\\ & = 2\cdot 3t^2\cdot \sin t \cdot 6t + \big (\big (3t^2\big )^2 - 2e^{2\sin t}\big ) \cdot \cos t\\ & = 36t^3\sin t + \big ( 9t^4 - 2e^{2\sin t}\big )\cos t\\ \end {align*}
Let \(f(u,v) = uv^2\)
where \(\hspace {0.5cm} u = 3x-y\hspace {0.5cm}, \hspace {0.5cm} v = x^2y\)
\[f(u,v) = f\big (u(x,y), v(x,y)\big ) = F(x,y)\]
\begin {align*} \frac {\partial F}{\partial x} & = \frac {\partial F}{\partial u}\cdot \frac {\partial u}{\partial x} + \frac {\partial F}{\partial v}\cdot \frac {\partial v}{\partial x}\\ & = v^2(3) + 2uv(2xy)\\ & = 3x^4y^2 + 2(x^2y)(3x - y) \cdot 2xy\\ & = 15x^4y^2 - 4x^3y^3\\ \end {align*}
\begin {align*} ||||_y\hspace {1cm} \frac {\partial F}{\partial y} & = \frac {\partial F}{\partial u}\cdot \frac {\partial u}{\partial y} +\frac {\partial F}{\partial v}\cdot \frac {\partial v}{\partial y}\\ & = v^2(-1) + 2uv\cdot x^2\\ & = -\big (x^2y\big )^2 + 2\big (3x -y\big )\big (x^2y\big )\cdot x^2\\ & = -x^4y^2 + \big (6x -2y\big )x^4y\\ & = 6x^5y - 3x^4y^2\\ \end {align*}
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