Step by step
The question
Question: find the derivative of ∫ from 1 to x of et dt.
ddx∫ from 1 to x 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 x in place of t
Put x in place of t.
So the derivative is eˣ.
ddx∫ from 1 to x of et dt = ex
Careful!
There is nothing to take away.
The derivative of the area is just f at x.
The bottom number 1 does not matter.
Careful: not et − f(1) ✗
3 Check
Let's check at x = 2.
Our answer gives 7.3891.
answer at x = 2: 7.3891
Now the slope of the area function F between x = 2 and 2.001, with a calculator: about 7.393.
The same!
Right!
F(2.001) − F(2)0.001 ≈ 7.393
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
Careful: not et − f(1) ✗