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Math Help - Integral and derivate

  1. #1
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    Integral and derivate

    We know that f(x)=
    x^2
    ∫g(t)dt
    0

    What is f'(x) and which conditions are necessary for g(t)?

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  2. #2
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    Quote Originally Posted by antero View Post
    We know that f(x)=
    x^2
    ∫g(t)dt
    0

    What is f'(x) and which conditions are necessary for g(t)?
    Let u = x^2. Then y = \int_0^u g(t) \, dt.

    From the chain rule: \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = g(u) \cdot 2x = 2x g(x^2).
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  3. #3
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    Quote Originally Posted by antero View Post
    We know that f(x)=
    x^2
    ∫g(t)dt
    0

    What is f'(x) and which conditions are necessary for g(t)?

    By the "Fundamental theorem of Calculus", the derivative of \int_a^u g(t) dt, with respect to u, is g(u). Here, u= x^2 so, by the chain rule, \frac{df}{dx}= \frac{df}{du}\frac{du}{dx} = g(x^2)(2x).

    The only "conditions" on g are that it be continuous.
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