Derivation of the integral Evaluate
  $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm  ds}}\sec(t^2)\,\mathrm dt$$


I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if this is correct and if not some suggestions?
It is 

$$ [ \tan(\int_0^{e^{x^2}}\cos sds)]' =  [ \tan(\sin e^{x^2})]'$$


   $$\sec^2(\sin e^{x^2})  \cdot \cos e^{x^2}  \cdot e^{x^2} \cdot  (2x)  $$


Can you proceed from here?
$$\frac{d}{dx}\int_0^{\int_0^{e^x}\cos(s)ds} \sec(t^2)dt=\sec(\sin^2e^x)\frac{d}{dx}{\sin^2e^x}=\sec(\sin^2e^x).(\sin2e^x)e^x$$