Step by step
The question
Question: find the derivative of ∫ from 0 to 3x of cos t dt.
ddx∫ from 0 to 3x 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 Put the top in place of t
The top is not just x: it is 3x.
Call it u.
Its slope is u′.
top: u = 3x, u′ = 3
The chain rule: put u in place of t, then multiply by u′.
ddx∫ from 0 to 3x of cos t dt = f(u) × u′
Put 3x in place of t, times 3.
= cos(3x) × 3
Careful!
Do not stop at f of u.
The top is 3x, so we must multiply by its slope, 3.
Careful: not cos(3x) ✗
Write the 3 in front: 3 cos(3x).
= 3 cos(3x)
3 Check
Let's check at x = 1.
Our answer gives −2.97, in radians.
answer at x = 1: −2.97
Now the slope of the area function F between x = 1 and 1.001, with a calculator: about −2.971.
The same!
Right!
F(1.001) − F(1)0.001 ≈ −2.971
Let's look back.
First: the rule.
Then: put the top in place of t.
Last: check.
Answer
= 3 cos(3x)