Step by step
The question
Question: find the derivative of ∫ from 1 to x of (t² − 4) dt.
ddx∫ from 1 to x of (t2 − 4) dt
1 The rule
The rule: the derivative of the area from a to x is the function inside the integral, 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 x² − 4.
ddx∫ from 1 to x of (t2 − 4) dt = x2 − 4
Careful!
The derivative of the area is just f at x.
The bottom number 1 does not matter.
Careful: the bottom number 1 does not matter
3 Check
Let's check at x = 2.
Our answer gives 0.
answer at x = 2: 0
Now the slope of the area function F, the integral itself, between x = 2 and 2.001, with a calculator: about 0.002.
The same!
Right!
F(2.001) − F(2)0.001 ≈ 0.002
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
ddx∫ from 1 to x of (t2 − 4) dt = x2 − 4