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One independent variable.

Theorem 5.16   Let f be a function of two variables (x,y), differentiable on an open domain $\mathcal{D}$. Suppose that x and y are functions of a single variable t, differentiable on an open interval $\mathcal{I}$, and such that for every $t \in \mathcal{I}, \;
( x(t),y(t) ) \in \mathcal{D}$.

Then f ( x(t),y(t) ) is a function of t, differentiable on $\mathcal{I}$ and we have:

\begin{displaymath}\frac {\partial f}{\partial t} = \frac {\partial f}{\partial ...
...frac {dx}{dt} +
\frac {\partial f}{\partial y} \frac {dy}{dt}.
\end{displaymath}


  
Figure 1: The chain rule for two intermediate variables.
\begin{figure}
\mbox{\epsfig{file=chain1.eps,height=4cm} }
\end{figure}

Remark 5.17   Actually (x(t), y(t)) is a parametrization of a curve in the plane. The chain rule enables us to compute the derivative of a function f defined on a plane curve.

Example 5.18   Let z=f(x,y)=x2+y3. Then $\frac {\partial f}{\partial x}=2x$ and $\frac {\partial f}{\partial y}=3y^2$.

Suppose now that $x=\cos 2t$ and $y= \cos t$. Then $\frac {dx}{dt}=-2 \sin 2t$ and $\frac {dy}{dt}=- \sin t$. It follows:
\begin{align*}\frac {\partial f}{\partial t} & = 2x \frac {dx}{dt} +3y^2 \frac {...
...t -3 \cos ^2 t \sin t \\
\quad &= -2 \sin 4t - 3 \cos ^2 t \sin t.
\end{align*}
A direct computation shows that, as $z=f(x,y)=\cos^2 2t + \cos^3 t$, the derivative $\frac {dz}{dt}$ is equal to $-4 \cos 2t \sin 2t -3 \cos ^2 t \sin t$.

Theorem 5.19   Let f be a function of three variables (x,y,z), differentiable on an open domain $\mathcal{D}$. Suppose that x, y and z are functions of a single variable t, differentiable on an open interval $\mathcal{I}$, and such that for every $t \in \mathcal{I}, \;
( x(t),y(t),z(t) ) \in \mathcal{D}$.

Then w=f ( x(t),y(t),z(t) ) is a function of t, differentiable on $\mathcal{I}$ and we have:

\begin{displaymath}\frac {\partial f}{\partial t} = \frac {\partial f}{\partial ...
...\frac {dy}{dt}
+ \frac {\partial f}{\partial z} \frac {dz}{dt}
\end{displaymath}


  
Figure 2: The chain rule for three intermediate variables.
\begin{figure}
\mbox{\epsfig{file=chain2.eps,height=4cm} }
\end{figure}

Remark 5.20   Actually (x(t), y(t), z(t)) is a parametrization of a curve in the 3-dimensional space. The chain rule enables us to compute the derivative of a function f defined on a space curve.


next up previous contents
Next: Two independent variables. Up: The chain rule. Previous: The chain rule.
Noah Dana-Picard
2001-05-30