Finding the derivative of a multivariable integral? Okay so I'm asked to find $\frac{df}{dt}$ of $f(t)=\int_{1}^{t^2}{\frac{1}{s}e^{s^2t}ds}$. Letting $g(s)=\frac{1}{s}e^{s^2t}$ I get:
$$f'(t)=G'(t^2) \cdot 2t - G'(1) \cdot 0
=g(t^2) \cdot 2t = \frac{2}{t}e^{t^5} $$
I'm not sure where I'm going wrong here but I'm pretty sure it's incorrect. 

In general, $\dfrac{d}{dt}\displaystyle\int_{h_1(t)}^{h_2(t)}g(s,t)\,ds = g(h_2(t),t)h_2'(t) - g(h_1(t),t)h_1'(t) + \int_{h_1(t)}^{h_2(t)}\dfrac{\partial g}{\partial t}(s,t)\,ds$.
You did correctly work out the first two terms $g(h_2(t),t)h_2'(t) - g(h_1(t),t)h_1'(t)$. 
However, you forgot the last term $\displaystyle\int_{h_1(t)}^{h_2(t)}\dfrac{\partial g}{\partial t}(s,t)\,ds$. 
Notice that the last term $\displaystyle\int_{h_1(t)}^{h_2(t)}\dfrac{\partial g}{\partial t}(s,t)\,ds$ is zero if the integrand $g(s,t)$ does not depend on the parameter $t$. However, in your case, the integrand does depend on $t$, so you can't drop this term. 
