Step by step
The question
Question: find the derivative of ∫ from 1 to x of 1/(t³ + 1) dt.
ddx∫ from 1 to x of 1t3 + 1 dt =
1 The rule
The rule: the derivative of the area from a to x is the function at x.
Just put x in place of t.
ddx∫ from a to x of f(t) dt = f(x)
2 Put x in place of t
Put x in place of t.
So the derivative is 1/(x³ + 1).
ddx∫ from 1 to x of 1t3 + 1 dt = 1x3 + 1
Careful!
There is nothing to take away.
The derivative of the area is just f at x.
The bottom number 1 does not matter.
Careful: not 1t3 + 1 − f(1) ✗
3 Check
Let's check at x = 2.
Our answer gives 0.1111.
answer at x = 2: 0.1111
Now the slope of the area function F between x = 2 and 2.001, with a calculator: about 0.111.
The same!
Right!
F(2.001) − F(2)0.001 ≈ 0.111
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
Careful: not 1t3 + 1 − f(1) ✗