Question

How do you use the fundamental theorem of calculus to find F'(x) given `F(x)=int 1/t^2dt` from [1,x]?

Answer 1

`F'(x)=1/x^2`

Explanation:

The first part of the Fundamental Theorem of Calculus tells us that if

`F(x)=int_a^xf(t)dt` where `a` is just a constant, then `F'(x)=f(x)`.

This makes sense -- `F(x)` is an integral, differentiating `F(x)` to get `F'(x)` should just give us what we originally integrated, IE, the integrand `F(t)` evaluated from a constant to `x`.

The most important thing to note is that the variable of the derivative and variable of the integrand are different. The integrand is written in terms of `t,` the integral and derivative in terms of `x.`

Here, we have

`F(x)=int_1^x1/t^2dt`

And we see `a=1` (the value of `a` is totally irrelevant to our final answer, noting it anyways), `f(t)=1/t^2.`

Thus,

`F'(x)=1/x^2`