6.1. Inverse Functions http://www.ck12.org
Solution:
Sincef′(x) = 15 x^4 + 2 >0 for allx∈R,f−^1 (x)is differentiable at all values ofx.To find the derivative off−^1 ,if
we letx=f(y),then
x=f(y) = 3 y^5 + 2 y+ 1.So
dx
dy=^15 y(^4) + 2
and
dy
dx=
1
dx/dy=1
15 y^4 + 2.Since we are unable to solve foryin terms ofx,we leave the answer above in terms ofy.Another way of solving
the problem is to use Implicit Differentiation:
Since
x= 3 y^5 + 2 y+ 1 ,differentiating implicitly,
d
dx[x] =d
dx[^3 y(^5) + 2 y+ 1 ],
1 = ( 15 y^4 + 2 )dydx.
Solving fordydxwe finally obtain
dy
dx=
1
15 y^4 + 2 ,which is the same result.
Review Questions
In problems #1 - 3, find the inverse function offand verify thatf◦f−^1 =f−^1 ◦f=x.
- f(x) = 3 x+ 1
- √^3 x