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$$F(x)=\int_{3-x}^{x^2}\frac{\sin(4xy)}{y}dy$$

I need to find $F' \left(\frac 3 2 \right)$

The integrand is continuous in $\left[\frac 3 2, \frac 9 4 \right]$ so I tried Leibniz with chain rule and it turned into an extremely long expression. Is there a simpler, more elegant way of calculating this? Thank you.

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    Make a variable substitution if it is too difficult as is. – 2017-01-14
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    How would a substitution help (I assume you mean of the limits of integration)? – 2017-01-14
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    It removes the integrands dependence on x. Set $t = 4xy \implies dt = 4x dy$ and change the limits etc. Then you end up with $$F = \int_{12x-4x^{2}}^{4x^{3}} \frac{\sin(t)}{t} dt$$ which you can easily apply the Leibniz rule to. – 2017-01-14

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