Step by step
The question
Question: find the derivative of ∫ from x to 1 of √(t⁴ + 1) dt.
ddx∫ from x to 1 of √t4 + 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.
ddx∫ from a to x of f(t) dt = f(x)
2 Swap the limits
x is at the bottom.
Swap the limits: swapping them flips the sign.
ddx∫ from x to 1 of √t4 + 1 dt = −ddx∫ from 1 to x of √t4 + 1 dt
Careful!
x is the bottom limit here.
Swapping the limits gives a minus sign.
Do not lose it.
Careful: not answer √t4 + 1 (no minus) ✗
3 Put x in place of t
Put x in place of t.
So the derivative is − √(x⁴ + 1).
= −√x4 + 1
4 Check
Let's check at x = 2.
Our answer gives −4.1231.
answer at x = 2: −4.1231
Now the slope of the area function F between x = 2 and 2.001, with a calculator: about −4.125.
The same!
Right!
F(2.001) − F(2)0.001 ≈ −4.125
Let's look back.
First: the rule.
Then: swap the limits.
Then: put x in place of t.
Last: check.
Answer
= −√x4 + 1