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# At a Glance - Thinking Backwards

The chain rule patterns also help us to think backwards, which will be useful for something called "integration by substitution".

We'll get to that later, but what we mean by this is that given a derivative, the chain rule patterns we've covered may help us find the original function. More precisely, given f ' (x) we can venture a guess as to what f (x) might be.

We'll explore this further through the examples and exercises.

#### Example 1

 What is a function whose derivative is 2xex2?

#### Exercise 1

Determine a function whose derivative is

•  -3sin x(cos x)2

#### Exercise 2

Determine a function whose derivative is

•  6(x3 + 2x + 1)5(3x2 + 2)

#### Exercise 3

Determine a function whose derivative is

• cos(x + 4)

#### Exercise 4

Determine a function whose derivative is

#### Exercise 5

Determine a function whose derivative is

• -6sin(6x)