$f\left(x\right)\=g\left(h\left(x\right)\right)\Rightarrow f\prime \left(x\right)\=g\prime \left(h\left(x\right)\right) h\prime \left(x\right)$

Table 4.3.1   Chain rule for single-variable composite function

Composite Function

Chain Rule

$F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\right)$

$F\prime \left(t\right)\=f_x\left(x\left(t\right)\,y\left(t\right)\right)x\prime \left(t\right)\+f_y\left(x\left(t\right)\,y\left(t\right)\right)y\prime \left(t\right)$

$G\left(x\,y\right)\=g\left(f\left(x\,y\right)\right)$

$G_x\left(x\,y\right)\=g\prime \left(f\left(x\,y\right)\right)f_x$ 

$G_y\left(x\,y\right)\=g\prime \left(f\left(x\,y\right)\right)f_y$

$F\left(s\,t\right)\=f\left(x\left(s\,t\right)\,y\left(s\,t\right)\right)$

$F_s\left(s\,t\right)\=f_x\left(x\left(s\,t\right)\,y\left(s\,t\right)\right)x_s\left(s\,t\right)\+f_y\left(x\left(s\,t\right)\,y\left(s\,t\right)\right)y_s\left(s\,t\right)$ 

$F_t\left(s\,t\right)\=f_x\left(x\left(s\,t\right)\,y\left(s\,t\right)\right)x_t\left(s\,t\right)\+f_y\left(x\left(s\,t\right)\,y\left(s\,t\right)\right)y_t\left(s\,t\right)$

$F\left(s\,t\right)\=f\left(x\left(s\,t\right)\,y\left(s\,t\right)\,z\left(s\,t\right)\right)$

$F_s\=f_x x_s\+f_y y_s\+f_z z_s$ 
$F_t\=f_x x_t\+f_y y_t\+f_z z_t$

$F\left(r\,s\,t\right)\=f\left(x\left(r\,s\,t\right)\,y\left(r\,s\,t\right)\right)$

$F_r\=f_x x_r\+f_y y_r\+f_z z_r$ 

$F_s\=f_x x_s\+f_y y_s\+f_z z_s$

$F_t\=f_x x_t\+f_y y_t\+f_z z_t$

Table 4.3.2   Composite functions and their derivatives by an appropriate form of the chain rule

Example 4.3.1

The composition of $f\left(x\,y\right)\=3-x^2-y^2$ with $x\left(t\right)\=t\,y\left(t\right)\=t^2$ forms the function $F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\right)$. Obtain $F\prime \left(t\right)$ by an appropriate form of the chain rule, and again by writing the rule for $F$ explicitly. Give a graphical interpretation of $F\left(t\right)$.

Example 4.3.2

The composition of  $f\left(x\,y\right)\=\sin \left(2x-3y\right)$, with $x\left(t\right)\=t\+1\/t$, $y\left(t\right)\=t-1\/t$ forms the function $F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\right)$. Obtain $F\prime \left(t\right)$ by an appropriate form of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.

Example 4.3.3

The composition of $f\left(x\,y\right)\=\ln \left(2y\+\sqrt{1-x^2}\right)$ with $x\left(t\right)\=t^2$, $y\left(t\right)\=\sin \left(t\right)$ forms the function $F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\right)$. Obtain $F\prime \left(t\right)$ by an appropriate form of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.

Example 4.3.4

The composition of $f\left(x\,y\,z\right)\=x{\mathit{\ⅇ}}^{2y}\cos \left(4z\right)$ with $x\left(t\right)\=\sqrt{1-t}$, $y\left(t\right)\=t$, $z\left(t\right)\=1-t^2$ forms the function $F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\,z\left(t\right)\right)$. Obtain $F\prime \left(t\right)$ by an appropriate form of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.

Example 4.3.5

The composition of $f\left(x\,y\right)\=x^{3}y$ with $x\=r \cos \left(t\right)\,y\=r \sin \left(t\right)$ forms the function $F\left(r\,t\right)\=f\left(x\left(r\,t\right)\,y\left(r\,t\right)\right)$. Obtain the partial derivatives $F_r$ and $F_t$ by appropriate forms of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.$$

Example 4.3.6

The composition of $f\left(x\,y\right)\=\ln \left(3x^2\+4y^2\right)$ with $x\left(r\,s\right)\=3 r\+2 s\,y\left(r\,s\right)\=5 r-7 s$ forms the function $F\left(r\,s\right)\=f\left(x\left(r\,s\right)\,y\left(r\,s\right)\right)$. Obtain the partial derivatives $F_r$ and $F_s$ by appropriate forms of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.

Example 4.3.7

The composition of $f\left(x\,y\,z\right)\=\sqrt{x^2\+y^2\+z^2}$ with $x\left(r\,s\right)\={\ⅇ}^r\cos \left(s\right)$, $y\left(r\,s\right)\={\ⅇ}^r\sin \left(s\right)$, $z\left(r\,s\right)\=rs$ forms the function $F\left(r\,s\right)\=f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)$. Obtain the partial derivatives $F_r$ and $F_s$ by appropriate forms of the chain rule, and again by writing the rule for $F$ explicitly. Show that the results agree.

Example 4.3.8

Let $F\left(x\,y\right)\=f\left(r\left(x\,y\right)\,s\left(x\,y\right)\right)$ be defined by the composition of $f\left(r\,s\right)$ with $r\=x\/y$, $s\=y\/x$, for any sufficiently well-behaved function $f\left(r\,s\right)$. Show that $x{F}_x\+y{F}_y\=0$.

Example 4.3.9

Let $F\left(x\,y\right)\=f\left(r\left(x\,y\right)\,s\left(x\,y\right)\right)$ be defined by the composition of $f\left(r\,s\right)$ with $r\=x^2-y^2$, $s\=y^2-x^2$, for any sufficiently well-behaved function $f\left(r\,s\right)$. Show that $y{F}_x\+x{F}_y\=0$.

Example 4.3.10

If $F\left(x\,y\right)$ is the composition of $G\left(u\,v\right)$ with $u\=x^2-y^2$, $v\=2 x y$, obtain $F_x$ and $F_y$ in terms of $G_u$ and $G_v$.

Example 4.3.11

If $u\=f\left(x-y\,y-x\right)$, show that $u_x\+u_y\=0$.

Example 4.3.12

If $u\=F\left(\frac{y-x}{x y}\,\frac{z-x}{x z}\right)$, show that $x^2 u_x\+y^2 u_y\+z^2 u_z\=0$.

Example 4.3.13

If $u\=x^3 F\left(y\/x\,z\/x\right)$, show that $x u_x\+y u_y\+z u_z\=3 u$.

Example 4.3.14

If $w\=f\left(x\,y\right)$ and $x\=r \cosh \left(s\right)$, $y\=r \sinh \left(s\right)$, obtain ${\left(w_r\right)}^2-{\left(w_s\/r\right)}^2$ in terms of $f_x$ and $f_y$.

Example 4.3.15

If $z\=y f\left(x^2-y^2\right)$, show that $y z_x\+x z_y\=x z\/y$.

Example 4.3.16

If $z\=x y\+x f\left(y\/x\right)$, show that $x z_x\+y z_y\=x y\+z$.

Example 4.3.17

If $u$ is a function of $r\=\sqrt{x^2\+y^2\+z^2}$, show that ${\left(u_x\right)}^2\+{\left(u_y\right)}^2\+{\left(u_z\right)}^2\={\left(\frac{\mathrm{du}}{\mathrm{dr}}\right)}^2$.

Example 4.3.18

If the equation $f\left(x\,y\,z\right)\=0$ implicitly defines $z\=z\left(x\,y\right)$, obtain $z_x$ and $z_y$.

Example 4.3.19

If the equation $z\=f\left(x\,y\,z\right)$ implicitly defines $z\=z\left(x\,y\right)$, obtain $z_x$ and $z_y$.

Example 4.3.20

If the equation $z\=f\left(x\,y\,z\right)$ implicitly defines $x\=x\left(y\,z\right)$, obtain $x_y$ and $x_z$.

Example 4.3.21

If the equation $f\left(x\,y\right)\=g\left(y\,z\right)$ implicitly defines $z\=z\left(x\,y\right)$, obtain $z_x$ and $z_y$.

Example 4.3.22

If the equation $x^2\+y^2\=f\left(x\,y\,z\right)$ implicitly defines $x\=x\left(y\,z\right)$, obtain $x_y$ and $x_z$.

Example 4.3.23

If the equation $f\left(x\,y\right)\=0$ implicitly defines $y\=y\left(x\right)$, obtain $y″\left(x\right)$.

Example 4.3.24

If the equation $f\left(x\,y\,z\right)\=0$ implicitly defines $z\=z\left(x\,y\right)$, obtain $z_{\mathrm{xy}}$.

Example 4.3.25

If the equation $x^2\+y^3\+z^4\+u^5\=1$ implicitly defines $u\=u\left(x\,y\,z\right)$, and the equation $x\+y^2\+z^3\=1$ implicitly defines $z\=z\left(x\,y\right)$, obtain $u_x$ and $u_y$.

Example 4.3.26

If the equation $u\=x^3\+y^3\+z^3\+u^3$ implicitly defines $u\=u\left(x\,y\,z\right)$ and the equation $z\=x^2\+y^2\+z^2$ implicitly defines $z\=z\left(x\,y\right)$, obtain $u_x$ and $u_y$.

Example 4.3.27

If the equation $u\=x\+y\+z\+e^u$ implicitly defines $u\=u\left(x\,y\,z\right)$ and the equation $z\=x\+y\+\cos \left(z\right)$ implicitly  defines $z\=z\left(x\,y\right)$, obtain $u_x$ and $u_y$.

Example 4.3.28

If the equation $f\left(x\,y\,z\,u\right)\=0$ implicitly defines $u\=u\left(x\,y\,z\right)$ and the equation $g\left(x\,y\,z\right)\=0$ implicitly  defines $z\=z\left(x\,y\right)$, obtain $u_x$ and $u_y$.

Example 4.3.29

The composition of $f\left(x\,y\right)$ with $x\=r \cos \left(t\right)$, $y\=r \sin \left(t\right)$ produces the function $F\left(r\,t\right)\=f\left(x\left(r\,t\right)\,y\left(r\,t\right)\right)$. Express $f_x$ and $f_y$ in terms of $F_r$ and $F_t$.

Example 4.3.30

The composition of $f\left(x\,y\right)$ with $x\=r \cos \left(t\right)$, $y\=r \sin \left(t\right)$ produces the function $F\left(r\,t\right)\=f\left(x\left(r\,t\right)\,y\left(r\,t\right)\right)$. Express $f_{\mathrm{xx}}\+f_{\mathrm{yy}}$ in terms of $F_r\,F_{\mathrm{rr}}$, and $F_{\mathrm{tt}}$.