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Math Help - Implicit Differentiation Help

  1. #1
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    Implicit Differentiation Help

    How can I find the derivative of this implicit relation?

    How do I find dG/dt for G=(bv)/(1-(v^2/c^2))^(1/2)
    *b and c are constants
    *v depends on t
    I would really appreciate if you explained it to me so I can learn how to solve such problems.
    Thanks!
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  2. #2
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    Re: Implicit Differentiation Help

    Differentiate both sides with respect to t. Since v is a function of t, you have to use the chain rule.
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  3. #3
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    Re: Implicit Differentiation Help

    How do I use the chain rule since I don't know how v is a function of t?
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  4. #4
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    Re: Implicit Differentiation Help

    Just use \frac{dv}{dt}.
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  5. #5
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    Re: Implicit Differentiation Help

    Quote Originally Posted by citcat View Post
    How can I find the derivative of this implicit relation?

    How do I find dG/dt for G=(bv)/(1-(v^2/c^2))^(1/2)
    *b and c are constants
    *v depends on t
    I would really appreciate if you explained it to me so I can learn how to solve such problems.
    Thanks!
    Since v is a function of t, this is a composition of functions, with your "inner" function of t being v(t), and your "outer" function of v being \displaystyle \begin{align*}  \frac{b\,v}{\sqrt{1 - \frac{v^2}{c^2}}} \end{align*}. So to differentiate this function with respect to t, you need to use the chain rule and differentiate both the inner and outer functions, then multiply them together.

    The derivative of the inner function is \displaystyle \begin{align*} \frac{dv}{dt} \end{align*}. I'll leave you to evaluate the derivative of the outer function.
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  6. #6
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    Re: Implicit Differentiation Help

    So do I end up with (b*dv/dt)/0.5(1-v^2/c^2)^-0.5?
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  7. #7
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    Lightbulb Re: Implicit Differentiation Help

    dG/dt=\frac{b v'(t)}{\left(1-\frac{v(t)^2}{c^2}\right)^{3/2}}
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  8. #8
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    Re: Implicit Differentiation Help

    How do you get (1-v(t)^2/c^2)^(3/2) in the denominator? Thanks!
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