## Quick Reference

The following rule that gives the derivative of the composition of two functions: If *h*(*x*)=(*f* ○ *g*)(*x*)=*f*(*g*(*x*) for all *x*, then *h*′(*x*)=*f*′(*g*(*x*)*g*′(*x*). For example, if *h*(*x*)=(*x*^{2}+1)^{3}, then *h*=*f* ○ *g*, where *f*(*x*)=*x*^{3} and *g*(*x*)=*x*^{2}+1. Then *f*′(*x*)=3*x*^{2} and *g*′(*x*)=2*x*. So *h*′(*x*)=3(*x*^{2}+1)^{2}2*x*=6*x*(*x*^{2}+1)^{2}. Another notation can be used: if *y*=*f*(*g*(*x*), write *y*=*f*(*u*), where *u*=*g*(*x*). Then the chain rule says that *dy*/*dx*=(*dy*/*du*)(*du*/*dx*). As an example of the use of this notation, suppose that *y*=(sin *x*)^{2}. Then *y*=*u*^{2}, where *u*=sin *x*. So *dy*/*du*=2*u* and *du*/*dx*=cos *x*, and hence *dy*/*dx*=2 sin *x* cos *x*.

*Subjects:*
Mathematics.