6.6 合成関数の偏微分法

1.

$\begin{figure}\includegraphics[width=5cm]{CALCFIG/ren6-6-1a.eps} \end{figure}$

(a)

$\displaystyle \frac{dz}{dt}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = 2x \cdot 2 + 2 \cdot 3t^2$
	$\displaystyle =$	$\displaystyle 8t + 6t^2$

(b)

$\displaystyle \frac{dz}{dt}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = 2x(-\sin{t}) + 2y\cos{t}$
	$\displaystyle =$	$\displaystyle -2\sin{t}\cos{t} + 2\sin{t}\cos{t} = 0$

(c)

$\displaystyle \frac{dz}{dt}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = (2x+y)(-\sin{t}) + (x+4y)\cos{t}$
	$\displaystyle =$	$\displaystyle (2\cos{t} + \sin{t})(-\sin{t}) + (\cos{t} + 4\sin{t})\cos{t}$
	$\displaystyle =$	$\displaystyle -2\sin{t}\cos{t} - \sin^{2}{t} + \cos^{2}{t} + 4\sin{t}\cos{t}$
	$\displaystyle =$	$\displaystyle \cos^{2}{t} - \sin^{2}{t} + 2\sin{t}\cos{t} = \cos{2t} + \sin{2t}$

(d)

$\displaystyle \frac{dz}{dt}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = 3x^{2}y^{2} \cdot 2t + 2x^{3}y \cdot 3t^{2}$
	$\displaystyle =$	$\displaystyle 3t^{4}t^{6}\cdot 2t + 2t^{6}t^{3}\cdot 3t^{3} = 6t^{11} + 6t^{11} = 12t^{11}$

2.

$\begin{figure}\includegraphics[width=5cm]{CALCFIG/ren6-6-2a.eps} \end{figure}$

(a)

$\displaystyle \frac{\partial z}{\partial u}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} =2x + 2y \cdot 2$
	$\displaystyle =$	$\displaystyle 2x + 4y = 2(u - 2v) + 4(2u+v) = 10u$

$\displaystyle \frac{\partial z}{\partial v}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial v} = 2x(-2) + 2y$
	$\displaystyle =$	$\displaystyle -4x + 2y = -4(u - 2v) + 2(2u+v) = 10v$

(b)

$\displaystyle \frac{\partial z}{\partial u}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} = 2x+y + (x+4y)v$
	$\displaystyle =$	$\displaystyle 2(u+v) + uv + (u+v + 4uv)v = 2u + 2v + 2uv + v^{2} + 4uv^{2}$

$\displaystyle \frac{\partial z}{\partial v}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial v} = 2x + y + (x + 4y)u$
	$\displaystyle =$	$\displaystyle 2(u+v) + uv + (u+v + 4uv)u = 2u + 2v + 2uv + u^{2} + 4u^{2}v$

(c)

$\begin{figure}\includegraphics[width=5cm]{CALCFIG/ren6-6-2c.eps} \end{figure}$

$\displaystyle \frac{\partial z}{\partial u}$

$\displaystyle =$

$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} = 2xy^{2} \cdot v = 2uv\cdot v^{4} \cdot v = 2uv^{6}$

$\displaystyle \frac{\partial z}{\partial v}$	$\displaystyle =$	$\displaystyle \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\frac{dy}{dv} = 2xy^{2}\cdot u + 2x^{2}y \cdot 2v$
	$\displaystyle =$	$\displaystyle 2uv(v^{2})^{2}\cdot u + 2(uv)^{2}v^{2} \cdot 2v = 2u^2v^5 + 4u^{2}v^5 = 6u^2 v^5$