Step by step
The question
Question: find the derivative of ∫ from 0 to x² of (t + 1) 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.
Why?
Look at the area under f from 0 to x.
Move x a tiny bit to the right.
The area grows by a thin strip.
Its height is f(x).
So the area grows at the rate f(x).
2 Put the top in place of t
The top is not just x: it is x².
Call it u.
Its slope is u′.
The chain rule: put u in place of t, then multiply by u′.
Put x² in place of t, times 2x.
Careful!
Do not stop at f of u.
The top is x², so we must multiply by its slope, 2x.
Multiply it out: 2 x³ + 2x.
3 Check
Let's check at x = 1.
Our answer gives 4.
Now the slope of the area function F between x = 1 and 1.001, with a calculator: about 4.004.
The same!
Right!
Let's look back.
First: the rule.
Then: put the top in place of t.
Last: check.