1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Find the derivative of the function using the chain rule

  1. Jul 22, 2012 #1
    1. Find the derivative of the function

    2. [itex]\left(y= x sin\sqrt{x}\right)[/itex]

    3. I started using the product rule and then proceeded to use the chain rule, but I am wondering if I should have used the chain rule twice rather than starting with the product rule. Since I know that x is the outer function, sinx is the middle function, and [itex]\sqrt{x}[/itex] is the inner function.
  2. jcsd
  3. Jul 22, 2012 #2


    User Avatar
    Homework Helper

    No, you had it right the first time! The function you wrote is a product of x and a function composition (sin √x). The latter function is not inside the function x! (An example of a function composition with an outer, middle, and inner function would be something like this: [itex]f(x) = \left( \sin \sqrt{x} \right)^2[/itex].)

    So take the product rule, and then the chain rule, as you said you originally done.
  4. Jul 22, 2012 #3
    Ok, thanks for the advice.
  5. Jul 22, 2012 #4
    The answer would be [itex]\frac{x}{2sinx^{1/2}}[/itex] * cosx + sinx[itex]^{1/2}[/itex]
  6. Jul 22, 2012 #5
    Try again! :smile:

    It looks like you probably implemented the product rule correctly, but your chain rule took a wrong turn I think...
  7. Jul 22, 2012 #6
    It would be [itex]\frac{x}{2cosx^{1/2}}[/itex] * cosx + sinx[itex]^{1/2}[/itex]
    Last edited by a moderator: Jul 22, 2012
  8. Jul 22, 2012 #7
    What is the definition of the chain rule?

    Eg, if y=f(g(x)), then y'=? in terms of f, f', g, g' ?
    Last edited by a moderator: Jul 22, 2012
  9. Jul 22, 2012 #8


    User Avatar
    Homework Helper

    You're messing up the itex tags. Just use one pair, like this:
    [itex]\frac{x}{2\cos x^{1/2}} \cdot \cos x + \sin x^{1/2}[/itex]
    Anyway, it's still wrong. You're not taking the derivative of [itex]\sin \sqrt{x}[/itex] correctly.
  10. Jul 22, 2012 #9
    It would be [itex]\frac{x*cosx^{1/2}}{2x^{1/2}} + sinx^{1/2}[/itex]
  11. Jul 22, 2012 #10
    That's the ticket! :smile:

    But to simplify further, you could cancel a sqrt(x) top and bottom from the first term.
  12. Jul 22, 2012 #11


    User Avatar
    Homework Helper

    Let's clean up the LaTeX a bit more. Use the \cdot tag instead of a "*" for multiplication. Also, when writing trig functions, put a "\" before them and a space before the variable, like \sin x.
    [itex]\frac{x \cdot \cos \sqrt{x}}{2\sqrt{x}} + \sin \sqrt{x}[/itex]
    Anyway, you're almost done. You can simplify a little more. Notice the "x" in the numerator and the "√x" in the denomiator?

    EDIT: this was posted before oay's previous post was edited. :wink:
  13. Jul 22, 2012 #12
    Haha! :smile:

    I honestly edited mine before I saw your post though... So it's a win-win! :smile:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook