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Math Help - Differential using chain Rule

  1. #1
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    Differential using chain Rule

    suppose that f:R-->R is continuous. using the chain rule, or otherwise, show that the function F defined by
    F(x) = \exp(\int_0^xf) is differentiable and find its derivative.

    Fundamental Theorem of calculus say that the derivative of \int_a^xf(t)dt = f(x) I tried to use this to solve the above but no vain.
    thanks for any help.
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  2. #2
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    You have the FTC wrong: it is \int_a^b f'(x)\, \mathrm{d}x = f(b) - f(a), or equivalently, \frac{\mathrm{d}}{\mathrm{d}x}\int_a^xf(s)\,\mathr  m{d}s = f(x).

    Does this help?
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  3. #3
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    sorry still can't solve it with exponential!
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  4. #4
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    Let h(x) = \int_0^x f. Then F(x) = \exp(h(x)). Applying the chain rule gives F'(x) = \exp(h(x))\cdot h'(x). The exponential is now out of the way and you only have to differentiate h. Use the FTC for this.

    You can do eet!
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  5. #5
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    Define the function  g(t) to be the antiderivative of  f(t)

    Then,  \int_{0}^{x} f(t) dt = g(x) - g(0)

    So,  F(x) = e^{g(x) - g(0)}
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