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!