How to differentiate this integral?

[jrsh]

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

 

[HallsofIvy]

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

 

[tacman]

for your question! To differentiate this integral, we can use the chain rule. The chain rule states that for a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x). In this case, our composite function is d(g(x)) and the derivative of F(x) will be given by f(x) * g'(x). However, since g(x) may or may not be differentiable, we need to consider two cases.

1. If g(x) is differentiable, then we can simply apply the chain rule as stated above to find the derivative of F(x). This will give us d(F(x)) = f(x) * g'(x).

2. If g(x) is not differentiable, then we need to use a different approach. We can rewrite the integral as F(x) = \int\limits_a^x f(x) d(u), where u = g(x). Then, we can use the fundamental theorem of calculus to find the derivative of F(x), which is given by F'(x) = f(x) * d(u)/dx. Finally, we can substitute back in u = g(x) to get d(F(x)) = f(x) * g'(x).

I hope this helps! Let me know if you have any further questions.