How to differentiate an integral?

[owlpride]

I just need a short reminder from Calculus. Suppose you have a linear functional \alpha from C1[-1,1] to \Re, given by

\alpha(f) = \int_{-1}^{1}f(t)g(t)dt

for some fixed continuous function g. What is \frac{d \alpha}{d f}?

 

[djeitnstine]

\frac{d \alpha}{d f} = \int_{-1}^{1}\frac{d}{df}f(t)g(t)dt

 

[owlpride]

Is there any way to simplify (or expand) this?

I am tempted to think that

\int_{-1}^{1}\frac{d}{df}f(t)g(t)dt = \int_{-1}^{1}g(t)dt

but that cannot be right because it's just a constant and integrating it would give me f times a constant instead of an integral involving f .

 

[djeitnstine]

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

check that out

 

[owlpride]

Thanks a lot!