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Homework Help: FTC with a two-variable function

  1. Feb 12, 2014 #1

    I don't understand where the last term comes from, the one that's an integral of a partial derivative. When I solve it using the FTC I get the same answer minus that term.

    If I differentiate first then integrate I get that term but then none of the others.
  2. jcsd
  3. Feb 12, 2014 #2

    Ray Vickson

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    Homework Helper

    Work it out for yourself from first principles: if
    [tex] F(x) = \int_{a(x)}^{b(x)} f(t,x) \, dt,[/tex]
    [tex] \frac{d}{dx} F(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} [/tex]
    Work out the value of the numerator for small ##h > 0##; you will see that in general it involves several types of terms, and these give limits that are the terms in the final result you want.
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