Differentiate $f(x)=\int_x^{10}e^{-xy^2}dy$ with respect to $x$ I am trying to find $f'(x)$ when $0\leq x\leq 10$. I know I could use the formula given on this wikipedia page: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign but I have been asked to justify all steps of the calculation so this isn't allowed.
I have been given a hint to let $I(a,b,c)=\int_a^bf(x,c)dx$ and then told to show that $f$ satisfies all conditions necessary for FTC1 and the theorem of differentiation of integrals depending on a parameter.
The problem I am having is translating $f(x)$ into something of the same form as $I(a,b,c)$. Can anyone help?
EDIT: I think I've done it now using the method described by @mvggz . Is this the final answer once the $u$ has been substituted back out:
$$ f'(x)=-\frac{1}{x} \int_x^{10} e^{-xy^2} dy + \frac{5}{x} e^{-100x}-\frac{3}{2}e^{-x^3}$$

The trick in this cases is writing the integral as a composition:
$$H(x,z)=-\int_{10}^z e^{-xy^2}dy$$
$$\Delta(x)=(x,x)$$
$$\int_x^{10} e^{-xy^2}dy=I(x)=H(x,x)=H(\Delta(x)).$$
The partial derivatives of $\Delta$ are obvious, the partial derivatives of $H$
 can be calculated using differentiation under the integral sign (x) and the Fundamental theorem of the Calculus (z) .
Now you can calculate $I'(x)$ using the chain rule.
