Finding a derivative for every X

Question

I need to find the derivative of this function for every x$\in$R. I tried breaking it down into 2 integrals and using the taylor series but had no luck. Any ideas?

F(x)=$\int$$_0^x$cos$\frac{1}{t-1}$dt

Answer 1

Hint: The fundamental theorem of calculus states that $$\frac d{dx}\left(\int_a^x f(t)\,\mathrm dt \right)= f(x)$$

Answer 2

This is just the Fundamental Theorem of Calculus:

$$\frac{d}{dx}\int_0^x \cos\frac{1}{t-1}dt = \cos\frac{1}{x-1}$$

Since you're doing Taylor series now, I suspect that you covered this a few months ago in your course; I'd just go back and review old notes.