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Math Help - Derivative of an Integral

  1. #1
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    Derivative of an Integral

    Hi everyone! I was wondering if anyone here could kindly point me in the direction of how to get around solving for the derivative of \int_x^{2x} 2^{t^2} \, dt w.r.t. x. Thank you very much in advance.

    Also, I'm new here, so if I'm flouting any rules please do let me know and I will gladly make changes as necessary. Thanks again!
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  2. #2
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    Quote Originally Posted by synapse View Post
    Hi everyone! I was wondering if anyone here could kindly point me in the direction of how to get around solving for the derivative of \int_x^{2x} 2^{t^2} \, dt w.r.t. x. Thank you very much in advance.

    Also, I'm new here, so if I'm flouting any rules please do let me know and I will gladly make changes as necessary. Thanks again!

    The fundamental theorem of integral calculus tells us that:

    \int\limits_x^{2x} 2^{t^2} \, dt\,=F(2x)-F(x)\,,\,\,with\,\,\,F'(x)=2^{x^2} and thus \frac{d}{dx}\int\limits_x^{2x} 2^{t^2} \, dt=2F'(2x)-F'(x) and there you go./

    Tonio
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  3. #3
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    There are two theorems we may use to our advantage here:

    \begin{aligned}<br />
\frac{d}{dx}\int_0^x f(t)\,dt&=f(x)\\<br />
\int_x^0 f(t)\,dt&=-\int_0^x f(t)\,dt.<br />
\end{aligned}

    The first implies that as x moves to the right, the area under a curve changes at a rate equal to the height of the function at x. The second is defined for convenience in calculus.
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