I am so new on Mathematica. I try to find the first variation of this function according to $t$ on mathematica but I could not achieve. Here is the function ;

$$\int_{0}^{\infty}u\left(c\left(t\right)\right)\exp\left\{ -\int_{0}^{t}\theta\left(c\left(s\right)\text{d}s\right)\right\} \text{d}t$$

How can I find the first variation of this integral according to $t$ ?

Thanks in advance.

  • $\begingroup$ Have you tried to read the documentation? The variable t is only a dummy variable, that expression is not a function of t. Therefore its derivative is zero. $\endgroup$ – rhermans Oct 11 '14 at 21:25

In general you write your integral using Integrate

Integrate[f[x], {x, a, t}]


And the derivatives using D

D[g[x], x]

To differentiate the integral you can write

D[Integrate[f[x], {x, a, t}], t]

In your case

D[Integrate[ u[c[t]] Exp[-Integrate[th[c[s]], {s, 0, t}]], {t, 0, \[Infinity]}], t]


| improve this answer | |
  • 5
    $\begingroup$ Sometimes it's useful to Inactivate Integrate so that you can perform the derivative without needlessly trying to evaluate a slow symbolic integral. $\endgroup$ – Carlo Jan 10 '15 at 11:28

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