What is the derivative, with respect to x, of ∫ from 0 to x of √(t^2… Step by Step

Step by step

The question

Question: find the derivative of ∫ from 0 to x of √(t² + 4) dt.

ddx∫ from 0 to x of √t² + 4 dt

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The rule

The rule: the derivative of the area from a to x is the function inside the integral, 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 √(x² + 4).

ddx∫ from 0 to x of √t² + 4 dt = √x² + 4

Careful!

The derivative of the area is just f at x.

The bottom number 0 does not matter.

Careful: the bottom number 0 does not matter

3 Check

Let's check at x = 1.

Our answer gives 2.2361.

answer at x = 1: 2.2361

Now the slope of the area function F, the integral itself, between x = 1 and 1.001, with a calculator: about 2.236.

The same!

Right!

F(1.001) − F(1)0.001 ≈ 2.236

4 Pick the choice

Our answer is √(x² + 4).

That is choice C.

C) √x² + 4

5 Why not the others

A is √(x² + 4) − 2.

Wrong: took away the value at 0: the derivative of the integral is just the inside at x.

A) √x² + 4 − 2 ✗ took away the value at 0: the derivative of the integral is just the inside at x

B is x/√(x² + 4).

Wrong: differentiated √(t² + 4) as well.

B) x/√x² + 4 ✗ differentiated √(t^2 + 4) as well

D is √(t² + 4).

Wrong: put x in place of t.

D) √t² + 4 ✗ put x in place of t

Let's look back.

First: the rule.

Then: put x in place of t.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer