Chain rule for functions of two variables

Summary

2 variables, 1 variable

• If $z=f\left( x,y \right)$ and $x=g(t)$ and $y=h\left( t \right)$ then we can create a composite function

$$z=F\left( t \right)=f\left( g\left( t \right),h\left( t \right) \right)$$

• Example: $z=f\left( x,y \right)=xe^y$ , $x=t^2$ , $y=4t+1$ then $z=F\left( t \right)=t^2e^{4t+1}$
• If all functions are smooth then

$$F'\left( t \right)=f'_x\left( x,y \right)x'\left( t \right)+f'_y\left( x,y \right)y'\left( t \right)$$

• Example (continued). $f'_x\left( x,y \right)=e^y=e^{4t+1}$ , $x'\left( t \right)=2t$ , $f'_y\left( x,y \right)=xe^y=t^2e^{4t+1}$ and $y'\left( t \right)=4$ . We have

$$F'\left( t \right)=e^{4t+1}2t+4t^2e^{4t+1}$$

• We can confirm that this is correct by differentiating $F\left( t \right)=t^2e^{4t+1}$ with respect to $t$ .

Slope of a level curve

• $z=f\left( x,y \right)=c$ for some constant $c$ is a level curve for $f\left( x,y \right)$ . It is also an implicit relationship between $x$ and $y$ and $y=y\left( x \right)$ .
• The composite function is $z=F\left( x \right)=f\left( x,y\left( x \right) \right)=c$
• Taking the derivative of both sides using the chain rule:

$$\frac{d}{dx}F\left( x \right)=f'_x\left( x,y \right)⋅1+f'_y\left( x,y \right)⋅y'=0$$

• or (assuming $f'_y\left( x,y \right)≠0$ )

$$y'=- \frac{f'_x\left( x,y \right)}{f'_y\left( x,y \right)}$$

2 variables, 2 variables

• If $z=f\left( x,y \right)$ and $x=g(s,t)$ and $y=h\left( s,t \right)$ then we can create a composite function

$$z=F\left( s,t \right)=f\left( g\left( s,t \right),h\left( s,t \right) \right)$$

• Example: $z=f\left( x,y \right)=xy^2$ , $x=s^2+t^2$ , $y=st$ then

$$z=F\left( s,t \right)=\left( s^2+t^2 \right)s^2t^2=s^4t^2+s^2t^4$$

• If all functions are smooth then

$$F'_s\left( s,t \right)=f'_x\left( x,y \right)x'_s\left( s,t \right)+f'_y\left( x,y \right)y'_s\left( s,t \right)$$

$$F'_t\left( s,t \right)=f'_x\left( x,y \right)x'_t\left( s,t \right)+f'_y\left( x,y \right)y'_t\left( s,t \right)$$

• Example (continued). $f'_x\left( x,y \right)=y^2=s^2t^2$ , $f'_y\left( x,y \right)=2xy=2\left( s^2+t^2 \right)st$ , $x'_s\left( s,t \right)=2s$ , $x'_t\left( s,t \right)=2t$ , $y'_s\left( s,t \right)=t$ , $y'_t\left( s,t \right)=s$ . We have

$$F'_s\left( s,t \right)=s^2t^22s+2\left( s^2+t^2 \right)stt=4s^3t^2+2st^4$$

$$F'_t\left( s,t \right)=s^2t^22t+2\left( s^2+t^2 \right)sts=4t^3s^2+2s^4t$$

• We can confirm that this is correct by differentiating $F\left( s,t \right)=s^4t^2+s^2t^4$ with respect to $s$ and $t$ .