Results 1 to 4 of 4

Math Help - Product Differentiation

  1. #1
    Newbie
    Joined
    Feb 2009
    From
    Phoenix
    Posts
    6

    Product Differentiation

    This is a homework problem on the Algebra of Derivatives section in my text. I'm not sure if I'm handling the f(x^3) part correctly. Any suggestions (or if it's right, let me know) would help ease my mind greatly.

    Problem
    f:R \rightarrow R is differentiable.

    g(x)=x^2 f(x^3)

    Prove g(x) is differentiable and find the derivative.

    Proof
    We know that f(x) is differentiable over the real numbers, and clearly, if x is a real number, then x^3 is a real number, so f(x^3) is differentiable. Let h(x)=x^2. Then h(x) is differentiable, and by the Algebra of Derivatives, g(x) must be differentiable.

    It follows from the product rule that g(x)=2xf(x^3)+x^2 f'(x^3).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,987
    Thanks
    1650
    Quote Originally Posted by Junesong View Post
    This is a homework problem on the Algebra of Derivatives section in my text. I'm not sure if I'm handling the f(x^3) part correctly. Any suggestions (or if it's right, let me know) would help ease my mind greatly.

    Problem
    f:R \rightarrow R is differentiable.

    g(x)=x^2 f(x^3)

    Prove g(x) is differentiable and find the derivative.

    Proof
    We know that f(x) is differentiable over the real numbers, and clearly, if x is a real number, then x^3 is a real number, so f(x^3) is differentiable. Let h(x)=x^2. Then h(x) is differentiable, and by the Algebra of Derivatives, g(x) must be differentiable.

    It follows from the product rule that g(x)=2xf(x^3)+x^2 f'(x^3).
    You have differentiated f(x^3) incorrectly. By the chain rule \frac{df(x^3)}{dx}= f'(x^3) \frac{d x^3}{dx}= 3x^2f'(x^3).
    That is what you must multiply by x^2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2009
    From
    Phoenix
    Posts
    6
    Quote Originally Posted by HallsofIvy View Post
    You have differentiated f(x^3) incorrectly. By the chain rule \frac{df(x^3)}{dx}= f'(x^3) \frac{d x^3}{dx}= 3x^2f'(x^3).
    That is what you must multiply by x^2.
    Then g(x)=2xf(x^3)+3x^4 f'(x^3)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,987
    Thanks
    1650
    Yes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differentiation using product rule
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 13th 2010, 02:52 AM
  2. Differentiation- Product Rule
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 11th 2010, 05:09 AM
  3. Help: Product Differentiation!
    Posted in the Calculus Forum
    Replies: 6
    Last Post: May 18th 2009, 11:06 AM
  4. Replies: 3
    Last Post: March 20th 2009, 03:46 AM
  5. Differentiation using product rule
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 21st 2007, 03:55 AM

Search Tags


/mathhelpforum @mathhelpforum