Step by step
The question
Question: find the derivative of ∫ from 2 to x of (3 t² − t) dt.
ddx∫ from 2 to x of (3t2 − t) 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 3 x² − x.
ddx∫ from 2 to x of (3t2 − t) dt = 3x2 − x
Careful!
There is nothing to take away.
The derivative of the area is just f at x.
The bottom number 2 does not matter.
Careful: not 3t2 − x − f(2) ✗
3 Check
Let's check at x = 3.
Our answer gives 24.
answer at x = 3: 24
Now the slope of the area function F between x = 3 and 3.001, with a calculator: about 24.009.
The same!
Right!
F(3.001) − F(3)0.001 ≈ 24.009
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
Careful: not 3t2 − x − f(2) ✗