Step by step
The question
Question: find the derivative of ∫ from x to 3 of (t² + 2) dt.
ddx∫ from x to 3 of (t2 + 2) 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 Swap the limits
x is at the bottom.
Swap the limits: swapping them flips the sign.
ddx∫ from x to 3 of (t2 + 2) dt = −ddx∫ from 3 to x of (t2 + 2) dt
Careful!
x is the bottom limit here.
Swapping the limits gives a minus sign.
Do not lose it.
Careful: not answer t2 + 2 (no minus) ✗
3 Put x in place of t
Put x in place of t.
So the derivative is −(x² + 2).
= −(x2 + 2)
4 Check
Let's check at x = 4.
Our answer gives −18.
answer at x = 4: −18
Now the slope of the area function F between x = 4 and 4.001, with a calculator: about −18.004.
The same!
Right!
F(4.001) − F(4)0.001 ≈ −18.004
Let's look back.
First: the rule.
Then: swap the limits.
Then: put x in place of t.
Last: check.
Answer
= −(x2 + 2)