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

1 Answer
Mar 22, 2018

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