Proof of Differentiation Rules

[Kristina123]

 

[Pedja]

Hint: Find the first derivative of [imath][p(x)]^2-[q(x)]^2[/imath]

 

[Kristina123]

Hint: Find the first derivative of [imath][p(x)]^2-[q(x)]^2[/imath]

Do I add in some kind of expression that equals zero but that also allows me to rearrange and factor things even further now? Or, would the 2 lead me to the constant I'm looking for?

 

[Pedja]

Substitute [imath]p'(x)[/imath] by [imath]q(x)[/imath] and [imath]q'(x)[/imath] by [imath]p(x)[/imath]. What can you conclude from the value of derivative?

 

[Kristina123]

Substitute [imath]p'(x)[/imath] by [imath]q(x)[/imath] and [imath]q'(x)[/imath] by [imath]p(x)[/imath]. What can you conclude from the value of derivative?

It equals zero? OH, so therefore there must be a constant!! But why must it equal a constant?

 

[Pedja]

It equals zero? OH, so therefore there must be a constant!! But why must it equal a constant?

See this proof.

 

[Kristina123]

See this proof.

I sort of don't understand how to interpret this proof. The math notations are throwing me off, but does all of this have to do with the Mean Value Theorem? I haven't learned this theorem yet

 

[Pedja]

Take for granted: If [imath]f'(x)=0[/imath] on [imath](a,b)[/imath] then [imath]f(x)[/imath] must be constant.