Step by step
The question
Question: find the derivative of ∫ from 2 to x of (t² − 9) dt.
ddx∫ from 2 to x of (t2 − 9) 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 − 9.
ddx∫ from 2 to x of (t2 − 9) dt = x2 − 9
Careful!
The derivative of the area is just f at x.
The bottom number 2 does not matter.
Careful: the bottom number 2 does not matter
3 Check
Let's check at x = 3.
Our answer gives 0.
answer at x = 3: 0
Now the slope of the area function F, the integral itself, between x = 3 and 3.001, with a calculator: about 0.003.
The same!
Right!
F(3.001) − F(3)0.001 ≈ 0.003
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
ddx∫ from 2 to x of (t2 − 9) dt = x2 − 9