1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Given f(x) = 4^√x - x + 3, find (f^-1)'(3).

  1. Jul 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the value of the derivative of the inverse of the function.

    Given that f(x) = (4^√x) - x + 3, find (f^-1)'(3).



    2. Relevant equations



    3. The attempt at a solution

    I know that to find the inverse derivative, one should switch the x and y's and then take the derivative, but that 4^√x is kind of throwing me off...
     
  2. jcsd
  3. Jul 22, 2011 #2

    Pyrrhus

    User Avatar
    Homework Helper

    You just need to compute [itex]\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} [/itex]

    You specify z= (4^√x), and then calculate dz/dx. You can do that by applying natural log as ln z = √x ln 4, take derivative, and then go back to compute the original dz/dx.

    You can do this because if h(t) = u(t) + w(t) and you know the differential operator has the linear property, and thus h' = u' + w', where the primes mean derivatives wrt to t.
     
    Last edited: Jul 22, 2011
  4. Jul 22, 2011 #3
    Pyrrhus, is right, but from experience many introductory calculus students won't understand everything he's saying.


    http://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx" [Broken]
    Keep in mind that site uses f(x)=f(x) sometimes and g(x) = [itex]f^{-1} (x)[/itex] other times.
    So context is essential for knowing if x is the horizontal, independent variable or x might better be thought of as the vertical, dependent variable. Alternately, the notation and concepts may be easier to understand if we ignore the usual interpretation of x vs. y.

    To evaluate g'(3) = [itex]f^{-1}[/itex]' we'll need to first find
    g(3). We need to apply f( g(3) ) = 3 to find g(3).
    So f(x) = 3.
    I'll let you work out x. For now, I'll call it b.
    Next step, find f'. You'll need.

    d/dx [itex]4^{\sqrt{x}}[/itex]
    First look up d/dt [itex]a^{t}[/itex] in a derivative table.
    Then apply chain rule.

    I trust you can find
    g'(3) = [itex]\frac{1}{f'(b)}[/itex]
    from there.

    Show us what that gives you or if you get the answer in the back of the book.
     
    Last edited by a moderator: May 5, 2017
  5. Jul 22, 2011 #4

    hunt_mat

    User Avatar
    Homework Helper

    In general though let [itex]y=f^{-1}(x)[/itex], apply the function f to both sides to obtain [itex]f(y)=x[/itex], differentiate with respect to x to obtain:
    [tex]
    f'(y)\frac{dy}{dx}=1
    [/tex]
    Divide by f'(y) to obtain:
    [tex]
    \frac{dy}{dx}=\frac{1}{f(y)}=\frac{1}{f(f^{-1}(x))}
    [/tex]
    With your problem, swap f and the inverse of f around, so [itex]y=f(x)[/itex] and [itex]x=f^{-1}(y)[/itex]
     
    Last edited: Jul 22, 2011
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Given f(x) = 4^√x - x + 3, find (f^-1)'(3).
Loading...