How to differentiate this integral?

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To differentiate the integral F(x) = ∫_a^x f(x) d(g(x)), it is essential to consider the differentiability of g(x). When g(x) is differentiable, the derivative F'(x) can be expressed as F'(x) = f(x)g'(x). However, if g(x) is not differentiable at certain points, F(x) will also be non-differentiable at those points. Therefore, the differentiability of F(x) directly depends on the behavior of g(x). Understanding these conditions is crucial for correctly applying differentiation techniques to this integral.
jrsh
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Hello,

I have an integral

F(x) = \int\limits_a^x f(x) d(g(x))

and g(x) may and may not be differentiable.

How can I differentiate d(F(x))?

Thanks
 
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At points where g is differentiable F'(x)= f(x)g'(x). At points where g is not differentiable F is not differentiable.
 

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