How do you use the Fundamental Theorem of Calculus to find the derivative of <mstyle displaystyl

Crystal Wheeler

Crystal Wheeler

Answered question

2022-07-01

How do you use the Fundamental Theorem of Calculus to find the derivative of 1 1 + t 2 d t from x to 5?

Answer & Explanation

Zackery Harvey

Zackery Harvey

Beginner2022-07-02Added 21 answers

d d x   x 5 1 1 + t 2   d t = - 1 1 + x 2
The FTOC tells us that:
d d x   a x   f ( t )   d t = f ( x ) for any constant a
(ie the derivative of an integral gives us the original function back).
We are asked to find:
E = d d x   x 5 1 1 + t 2   d t
(notice the upper bounds of the first integral are not in the correct format for the FTOC to be applied, directly). We can manipulate the definite integral using integral properties:
E = d d x   - 5 x 1 1 + t 2   d t
    = - d d x   5 x 1 1 + t 2   d t
We can now apply the FTOC to get;
E = - 1 1 + x 2

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?