8.
- Ify= cscx, findy" (π/6).
- Use the definitionof the derivativeto provethat
Answers
1.y= x cos x + sin x
3.y= 2 cos x sin x
7.y= 1
8.y= sec^2 x
9.y(π/6)= 14
The ChainRule
LearningObjectives
A studentwill be able to:
- Knowthe chainrule and its proof.
- Applythe chainrule to the calculationof the derivativeof a varietyof compositefunctions.
We want to derivea rule for the derivativeof a compositefunctionof the formf gin termsof the derivatives
offandg. This rule allowsus to differentiatecomplicatedfunctionsin termsof knownderivativesof simpler
functions.
The ChainRuleIfgis a differentiablefunctionatxandfis differentiableatg(x),
then the compositionfunctionf g=f(g(x)) is differentiableatx. The derivative
of the compositefunctionis:
(f g)'(x) =f'(g(x))g'(x). Anotherway of expressing,ifu=u(x) andf=f(u),
then