Calculus

• 《高等数学》同济版 全程教学视频（宋浩老师）

一元复合函数的求导法则

定理 1

如果函数 \(u=g(x)\) 在点 \(x\) 可导，而函数 \(y=f(u)\) 在对应点 \(u=g(x)\) 可导，那么复合函数 \(y=f(g(x))\) 在点 \(x\) 可导，且有

\[\frac{\mathrm{d} y}{\mathrm{d} x}={f}'(u) \cdot {g}'(x) \quad or \quad \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} u}\cdot \frac{\mathrm{d} u}{\mathrm{d} x}\]

严格来说函数 \(y=f(u)\) 实际上只是 \(u\) 的函数，并不是 \(x\) 的函数。 而 \(h(x)=(f\circ g)(x)\) 才是 \(x\) 的函数。这种记号上的随意会在复杂问题上造成困扰。

\[\begin{split}\begin{align} \cfrac{h(x+\triangle x)-h(x)}{\triangle x } &=\cfrac{f(g(x+\triangle x))-f(g(x))}{\triangle x }\\ &=\cfrac{[f(g(x+\triangle x))-f(g(x))]}{[g(x+\triangle x)-g(x)]}\cfrac{[g(x+\triangle x)-g(x)]}{\triangle x } \end{align}\end{split}\]

\[\begin{split}\begin{align} \lim_{\triangle x \to 0}\cfrac{h(x+\triangle x)-h(x)}{\triangle x } &=\lim_{\triangle x \to 0}\cfrac{f(g(x+\triangle x))-f(g(x))}{\triangle x }\\ &=\lim_{\triangle x \to 0}\cfrac{[f(g(x+\triangle x))-f(g(x))]}{[g(x+\triangle x)-g(x)]}\cfrac{[g(x+\triangle x)-g(x)]}{\triangle x }\\ &=\frac{\mathrm{d} f(u)}{\mathrm{d} u} \Bigg|_{u=g(x)} \cdot \frac{\mathrm{d} g(x)}{\mathrm{d} x} \Bigg|_{x=x} \end{align}\end{split}\]

也就是说，如果把函数 \(y\) 定义成 \(y=f(u)\) ，那么复合函数就不要记作 \(y\) ，这有歧义。这里不妨记作 \(h\) , 一般来说，\(h\) 和 \(f\) 的表达式显然不是一回事，这是两个不同的函数。 一般有：

\[\begin{align} y_{0}=f(u_{0})= f(u(x_{0} ))= f(u(x ))\Bigg|_{x=x_{0}} \end{align}\]

举个例子：

\[\begin{split}\begin{align} y(u)=f(u)=2u\\ u(x)=g(x)=sin(x)\\ h(x)=f(g(x))=(f \circ g)(x)=2sin(x) \end{align}\end{split}\]

显然，这里的函数 \(y\) 和 函数 \(h\) 不是一回事。 我们可以将变量换为任何记号，比如 \(y(u)=f(u)=2u\) 或者 \(y(t)=f(t)=2t\) 或者 \(y(x)=f(x)=2x\) 对于函数 \(h\) 有 \(h(x)=2sin(x)\) ,如果此时也记做 \(y(x)=2sin(x)\) 则会造成歧义。

\[\begin{split}\begin{align} y(u)=f(u)=2u\\ u(x)=g(x)=sin(x)\\ \hat{y}(x)=h(x)=f(g(x))=(f \circ g)(x)=2sin(x) \end{align}\end{split}\]

这里的函数 \(y\) 和 函数 \(\hat{y}\) 应该加以区分。

\[y(sin(x))=f(sin(x))=\hat{y}(x)=2sin(x)\]

从这个角度来说，定理更合理的表述应该是：

如果函数 \(u=g(x)\) 在点 \(x\) 可导，而函数 \(y=f(u)\) 在对应点 \(u=g(x)\) 可导，那么复合函数 \(\hat{y}=f(g(x))\) 在点 \(x\) 可导，且有

\[\frac{\mathrm{d} \hat{y}(x)}{\mathrm{d} x}={f}'(u) \cdot {g}'(x) \quad or \quad \frac{\mathrm{d} \hat{y}(x)}{\mathrm{d} x}=\frac{\mathrm{d} y(u)}{\mathrm{d} u}\cdot \frac{\mathrm{d} u(x)}{\mathrm{d} x}\]

多元复合函数的求导法则

定理 1

如果函数 \(u=\phi(t)\) 及 \(v=\psi(t)\) 都在点 \(t\) 可导，函数 \(z=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(z=f(\phi(t),\psi(t))\) 在点 \(t\) 可导，且有

\[\frac{\mathrm{d} z}{\mathrm{d} t}=\frac{\partial z}{\partial u} \frac{\mathrm{d} u}{\mathrm{d} t}+ \frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{d} t}\]

更准确的叙述： 如果函数 \(u=\phi(t)\) 及 \(v=\psi(t)\) 都在点 \(t\) 可导，函数 \(z(u,v)=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(\hat{z}(t)=f(\phi(t),\psi(t))\) 在点 \(t\) 可导，且有

\[\frac{\mathrm{d} \hat{z}(t)}{\mathrm{d} t}=\frac{\partial z(u,v)}{\partial u} \frac{\mathrm{d} u(t)}{\mathrm{d} t}+ \frac{\partial z(u,v)}{\partial v} \frac{\mathrm{d} v(t)}{\mathrm{d} t}\]

即：

\[\frac{\mathrm{d} \hat{z}(t)}{\mathrm{d} t}=\frac{\partial z(u,v)}{\partial u} \frac{\mathrm{d} u(t)}{\mathrm{d} t}+ \frac{\partial z(u,v)}{\partial v} \frac{\mathrm{d} v(t)}{\mathrm{d} t}\]

定理 2

如果函数 \(u=\phi(x,y)\) 及 \(v=\psi(x,y)\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(z=f(\phi(x,y),\psi(x,y))\) 在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial z}{\partial x} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}\\ \frac{\partial z}{\partial y} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v} \frac{\partial v}{\partial y}\\ \end{align}\end{split}\]

类似的，设 \(u=\phi(x,y)\) 、\(v=\psi(x,y)\) 及 \(w=\omega(x,y)\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z=f(u,v,w)\) 在对应点 \((u,v,w)\) 具有连续偏导数，那么复合函数

\[z=f(\phi(x,y),\psi(x,y),\omega(x,y))\]

在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial z}{\partial x} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial x}\\ \frac{\partial z}{\partial y} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v} \frac{\partial v}{\partial y}+ \frac{\partial z}{\partial w} \frac{\partial w}{\partial y} \end{align}\end{split}\]

更准确的叙述： 如果函数 \(u=\phi(x,y)\) 及 \(v=\psi(x,y)\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z(u,v)=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(\hat{z}(x,y)=f(\phi(x,y),\psi(x,y))\) 在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v)}{\partial v} \frac{\partial v(x,y)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v)}{\partial v} \frac{\partial v(x,y)}{\partial y}\\ \end{align}\end{split}\]

类似的，设 \(u=\phi(x,y)\) 、\(v=\psi(x,y)\) 及 \(w=\omega(x,y)\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z(u,v,w)=f(u,v,w)\) 在对应点 \((u,v,w)\) 具有连续偏导数，那么复合函数

\[\hat{z}(x,y)=f(\phi(x,y),\psi(x,y),\omega(x,y))\]

在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial v} \frac{\partial v(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial w} \frac{\partial w(x,y)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial v} \frac{\partial v(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial w} \frac{\partial w(x,y)}{\partial y} \end{align}\end{split}\]

定理 3

如果函数 \(u=\phi(x,y)\) 在点 \((x,y)\) 具有对 \(x\) 及对 \(y`的偏导数，函数 :math:`v=\psi(y)\) 在点 \(y\) 可导，函数 \(z=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(z=f(\phi(x,y),\psi(y))\) 在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial z}{\partial x} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}\\ \frac{\partial z}{\partial y} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{d} y}\\ \end{align}\end{split}\]

如果复合函数的某些中间变量本身又是复合函数的自变量 设 \(u=\phi(x,y)\) 、\(v=x\) 及 \(w=y\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z=f(u,v,w)\) 即 \(z=f(u,x,y)\) 在对应点 \((u,v,w)\) 具有连续偏导数，那么复合函数

\[z=f(\phi(x,y),x,y)\]

有

\[\begin{split}\begin{align} \frac{\partial v}{\partial x}&=1, \quad \frac{\partial v}{\partial y}=0\\ \frac{\partial w}{\partial x}&=0, \quad \frac{\partial w}{\partial y}=1 \end{align}\end{split}\]

继续，有

\[\begin{split}\begin{align} \frac{\partial z}{\partial x} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v} \cdot 1+ \frac{\partial z}{\partial w} \cdot 0\\ \frac{\partial z}{\partial y} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v} \cdot 0+ \frac{\partial z}{\partial w} \cdot 1 \end{align}\end{split}\]

继续，有

\[\begin{split}\begin{align} \frac{\partial z}{\partial x} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v}\\ \frac{\partial z}{\partial y} & = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+ \frac{\partial z}{\partial w} \end{align}\end{split}\]

更准确的叙述： 如果函数 \(u=\phi(x,y)\) 在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(v=\psi(y)\) 在点 \(y\) 可导，函数 \(z(u,v)=f(u,v)\) 在对应点 \((u,v)\) 具有连续偏导数，那么复合函数 \(\hat{z}(x,y)=f(\phi(x,y),\psi(y))=\hat{f}(x,y)\) 在点 \((x,y)\) 的两个偏导数都存在，且有

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v)}{\partial v} \cancelto{0}{\frac{\partial v(x,y)}{\partial x}}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v)}{\partial v} \cancelto{\frac{\mathrm{d} v(y)}{\mathrm{d} y}}{\frac{\partial v(x,y)}{\partial y}} \end{align}\end{split}\]

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v)}{\partial v} {\frac{\mathrm{d} v(y)}{\mathrm{d} y}} \end{align}\end{split}\]

如果复合函数的某些中间变量本身又是复合函数的自变量 设 \(u=\phi(x,y)\) 、\(v=x\) 及 \(w=y\) 都在点 \((x,y)\) 具有对 \(x\) 及对 \(y\) 的偏导数，函数 \(z(u,v,w)=f(u,v,w)\) 即 \(z=f(u,x,y)\) 在对应点 \((u,v,w)\) 具有连续偏导数，那么复合函数

\[\hat{z}(x,y)=f(\phi(x,y),x,y)=\hat{f}(x,y)\]

有：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial v} \frac{\partial v(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial w} \frac{\partial w(x,y)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial v} \frac{\partial v(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial w} \frac{\partial w(x,y)}{\partial y} \end{align}\end{split}\]

继续，有：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial v} \cancelto{1}{\frac{\partial v(x,y)}{\partial x}}+ \frac{\partial z(u,v,w)}{\partial w} \cancelto{0}{\frac{\partial w(x,y)}{\partial x}}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial v} \cancelto{0}{\frac{\partial v(x,y)}{\partial y}}+ \frac{\partial z(u,v,w)}{\partial w} \cancelto{1}{\frac{\partial w(x,y)}{\partial y}} \end{align}\end{split}\]

即：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial v}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial w} \end{align}\end{split}\]

即：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial z(u,v,w)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial z(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial z(u,v,w)}{\partial y} \end{align}\end{split}\]

也可以写成：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial f(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial f(u,v,w)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial f(u,v,w)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial f(u,v,w)}{\partial y} \end{align}\end{split}\]

继续，也可以写成：

\[\begin{split}\begin{align} \frac{\partial \hat{z}(x,y)}{\partial x} & = \frac{\partial f(u,x,y)}{\partial u} \frac{\partial u(x,y)}{\partial x}+ \frac{\partial f(u,x,y)}{\partial x}\\ \frac{\partial \hat{z}(x,y)}{\partial y} & = \frac{\partial f(u,x,y)}{\partial u} \frac{\partial u(x,y)}{\partial y}+ \frac{\partial f(u,x,y)}{\partial y} \end{align}\end{split}\]

需要注意的是，虽然数值上有：

\[\begin{split}\begin{align} \hat{z}(x,y)&= \hat{f}(x,y)=z(u(x,y),x,y)=f(u(x,y),x,y)\\ \end{align}\end{split}\]

但是，函数形式上

\[\begin{split}\begin{align} \hat{z}&\ne z\\ \hat{f}&\ne f\\ \hat{z}&\equiv \hat{f}\\ {z}&\equiv {f}\\ \end{align}\end{split}\]