Step by step
The question
Question: find the derivative of ∫ from x to 5 of cos t dt.
ddx∫ from x to 5 of cos t 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 5 of cos t dt = −ddx∫ from 5 to x of cos t dt
Careful!
x is the bottom limit here.
Swapping the limits gives a minus sign.
Do not lose it.
Careful: not answer cos x (no minus) ✗
3 Put x in place of t
Put x in place of t.
So the derivative is −cos x.
= −cos x
4 Check
Let's check at x = 6.
Our answer gives −0.9602, in radians.
answer at x = 6: −0.9602
Now the slope of the area function F between x = 6 and 6.001, with a calculator: about −0.96.
The same!
Right!
F(6.001) − F(6)0.001 ≈ −0.96
Let's look back.
First: the rule.
Then: swap the limits.
Then: put x in place of t.
Last: check.
Answer
= −cos x