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 \(t\) .