You can fix the problem that the lower limit of integration is a variable by breaking it into to integrals:
where "a" is any fixed number- the lower limit won't be relevant to the derivative.
To fix the problem that the upper limit is not just "x", make a change of variable. If we let , then when [tex]t= x^7[tex], u will be equal to x. , of course, and so the first integral becomes
.
For the second integral let so that when , . and so the integral becomes
That is,
and now you can easily apply the Fundamental Theorem of Calculus to the two integrals on the right. The derivative with respect to x is, modulo any careless errors,
In fact, that can be generalized to "Leibniz' formula":
which is, I suspect, how FernandoRevilla got his answer so quickly!
I got it in the following way:
Using the Fundamental Theorem of Calculus and Chain's Rule we inmediately obtain:
(of course differentiable, etc)
Fernando Revilla