### Chain Rule & Integration by Substitution

Package of problems

## Things you might have noticed

Look at the functions in the table below.

Is there a property or feature that is shared by two functions in a row but not the third? If so, we can say that the third function is the odd one out.

Find a reason why each function in the table could be the odd one out in its row.

 $y=\sin x^2$ $y=\ln{x^2}$ $y=\tan x (\sec^2 x-1)$ $y=9x^2-6x+1$ $y=\ln{3x}$ $y=\sqrt{3x-1}$ $y=e^{5x}$ $y=\dfrac{1}{x^2+4x+4}$ $y=e^{x+4}$

Here are some reasons why each function is the odd one out. Can you think of others?

Which functions could you have identified as an odd one out from the formula? When did the graph help? How much extra information did finding $\dfrac{dy}{dx}$ give you?

What features or properties does $y$ inherit from the functions it’s composed from?

Can you also find an odd one out within each column?

Some of the ideas above can also explain why a function is the odd one out in its column, but there are other reasons too.