Find the derivative of
f(x)=xx2+1.
Since (x2+1)′=2x and (x)′=1,
f′(x)=x2x(2x)−(1)(x2+1)=x22x2−x2−1=x2x2−1. □
Find the derivative of
f(x)=x2ex.
Since (ex)′=ex and (x2)′=2x,
f′(x)=x4(x2)(ex)−(2x)(ex)=x4x2ex−2xex=x3xex−2ex. □
If f(x)=x3sinx, what is f′(x)?
Since (sinx)′=cosx and (x3)′=3x2,
f′(x)=x6(x3)(cosx)−(sinx)(3x2)=x6x2(xcosx−3sinx)=x4xcosx−3sinx. □
Some problems call for the combined use of differentiation rules:
Find the derivative of
f(x)=e3xecosx+tanx.
Since (ecosx+tanx)′=−sinxecosx+sec2x and (e3x)′=3e3x,
f′(x)=(e3x)2(e3x)(−sinxecosx+sec2x)−(3e3x)(ecosx+tanx)=e3x(−sinxecosx+sec2x)−3(ecosx+tanx)=e3x−sinxecosx+sec2x−3ecosx−3tanx. □
If that last example was confusing, visit the page on the chain rule.