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Compute \(f'\left(0\right)\). $$\large f\left(x\right) = \int\limits_{\cos x}^{\sin x} e^{t^2+xt}dt.$$

Mathematics
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Leibnitz rullllllleeeeeeeeeeeeeeeeee!!!!!!!!
\[\frac{\int\limits_{h(x)}^{g(x)}f(t).dt}{dx}= f(g(x)).g'(x) - f(h(x)).h'(x)\] Here, \(f(t) = e^{t^2 + xt}, g(x)=\sin x\text{ and }h(x)=\cos x\)
\[f'(x) = e^{\sin^2 x + x\sin x}.\cos x + e^{\cos^2 x + x\cos x}.\sin x \] \[SOoooooOOOOoooooo,~ f'(0) = e^0+ 0 = \boxed{1}\]

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(i.e., if I did not make a characteristically stupid error again -_-)
Read the question again apoorv. $$\large e^{t^2 + xt}$$
I am sorry, but isn't that 'x' supposed to be taken as a constant when you integrate wrt to 't'?? I can't figure out where am going wrong. Or may be I do..
No, you are differentiating with respect to \(x\).
Well that's the later part of the thing, but when am integrating, 'x' is a constant wrt 't', no balls with that. After am done the integration thingy, I am supposed to put in limits, and BAZINGA! the function is now in terms of 'x' completely. NOW we differentiate. We could follow all these steps, except that it would take ages for me to integrate that thing. So, Leibnitz to the rescue. Either this, or am missing your whole point.

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