How to Differentiate an Integral with a Variable Upper Limit?

[dbb04]

I have this equation


$$\int_t^T n(s) (1- e^{-c (T-s)}) ds = c F(T)$$


and I need to differentiate both sides with respect to T

$$\frac{\partial }{\partial T}$$

to get the following result

$$\int_t^T n(s) ( e^{-c (T-s)}) ds = \frac{\partial F(T)}{\partial T}$$

How was it done ? What integration and differentiation rule was used ? If you could show it step by step I would appreciate.

 

[shmoe]

I would rewrite the integral:

$$\int_t^T n(s) (1- e^{-c (T-s)}) ds = \int_t^T n(s)ds-e^{-cT}\int_t^T n(s) e^{cs}} ds$$

Then use the tried and true fundamental theorem of calculus (assuming g is continuous):

$$\frac{d}{dT}\int_{a}^{T}g(s)ds=g(T)$$

The purpose of rewriting was to remove any potentially confusing dependence of T from the integrands.

 

[dbb04]

Yeah, sure. Now I see it.

Thanks very much for the prompt reply