Step by step
The question
Question: find the derivative of ∫ from 0 to 2x of et dt.
ddx∫ from 0 to 2x of et 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 2x.
Call it u.
Its slope is u′.
top: u = 2x, u′ = 2
The chain rule: put u in place of t, then multiply by u′.
ddx∫ from 0 to 2x of et dt = f(u) × u′
Put 2x in place of t, times 2.
= e2x × 2
Careful!
Do not stop at f of u.
The top is 2x, so we must multiply by its slope, 2.
Careful: not e2x ✗
Write the 2 in front: 2 e²ˣ.
= 2 e2x
3 Check
Let's check at x = 1.
Our answer gives 14.7781.
answer at x = 1: 14.7781
Now the slope of the area function F between x = 1 and 1.001, with a calculator: about 14.793.
The same!
Right!
F(1.001) − F(1)0.001 ≈ 14.793
Let's look back.
First: the rule.
Then: put the top in place of t.
Last: check.
Answer
= 2 e2x