Step by step
The question
Question: find the derivative of ∫ from 1 to x² of t² dt.
ddx∫ from 1 to x² of t2 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 the top in place of t
The top is not just x: it is x².
Call it u.
Its slope is u′.
top: u = x2, u′ = 2x
The chain rule: put u in place of t, then multiply by u′.
ddx∫ from 1 to x² of t2 dt = f(u) × u′
Put x² in place of t, times 2x.
= x4 × 2x
Careful!
Do not stop at f of u.
The top is x², so we must multiply by its slope, 2x.
Careful: not x4 ✗
Multiply it out: 2 x⁵.
= 2x5
3 Check
Let's check at x = 2.
Our answer gives 64.
answer at x = 2: 64
Now the slope of the area function F between x = 2 and 2.001, with a calculator: about 64.08.
The same!
Right!
F(2.001) − F(2)0.001 ≈ 64.08
Let's look back.
First: the rule.
Then: put the top in place of t.
Last: check.
Answer
= 2x5