Step by step
The question
Question: find the derivative of ∫ from 0 to x of √(1 + t³) dt.
ddx∫ from 0 to x of √1 + 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)
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 x in place of t
Put x in place of t.
So the derivative is √(1 + x³).
ddx∫ from 0 to x of √1 + t3 dt = √1 + 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 √1 + t3 − f(0) ✗
3 Check
Let's check at x = 1.
Our answer gives 1.4142.
answer at x = 1: 1.4142
Now the slope of the area function F between x = 1 and 1.001, with a calculator: about 1.415.
The same!
Right!
F(1.001) − F(1)0.001 ≈ 1.415
Let's look back.
First: the rule.
Then: put x in place of t.
Last: check.
Answer
Careful: not √1 + t3 − f(0) ✗