Step by step
The question
Question: find the derivative of ∫ from 0 to x of t³ dt.
ddx∫ from 0 to x of t3 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 x³.
ddx∫ from 0 to x of t3 dt = x3
Careful!
There is nothing to take away.
The derivative of the area is just f at x.
The bottom number 0 does not matter.
Careful: not t3 − f(0) ✗
3 Check
Let's check at x = 1.
Our answer gives 1.
answer at x = 1: 1
Now the slope of the area function F between x = 1 and 1.001, with a calculator: about 1.002.
The same!
Right!
F(1.001) − F(1)0.001 ≈ 1.002
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
Careful: not t3 − f(0) ✗