A question on these derivative laws

[The_Z_Factor]

In the book I'm reading it says here that the derivative of a constant times a function equals the constant times the derivative of the function.

It gives the equation f(x)=c*u(x), then f '(x)=c*u'(x) Where f(x) and u(x) are functions of x.

So if f(x)=5x^2, that's 10x. I understand how to do these problems, they're easy, but then my book says

If f(x)=cx^n, then f '(x)=cnx^n-1

It says this example is a special case of a constant times a function.

I'm confused here. For whatever reason I'm thinking that these two would equal the same thing if used with the same variables? Could somebody tell me what my book means when it says this?

 

[arildno]

A power function is a special case of differentiable functions.

For ANY differentiable function f, the derivative of c*f is c*f', whether or not f is a power function.

 

[HallsofIvy]

In the book I'm reading it says here that the derivative of a constant times a function equals the constant times the derivative of the function.

It gives the equation f(x)=c*u(x), then f '(x)=c*u'(x) Where f(x) and u(x) are functions of x.

So if f(x)=5x^2, that's 10x. I understand how to do these problems, they're easy, but then my book says

If f(x)=cx^n, then f '(x)=cnx^n-1

It says this example is a special case of a constant times a function.

I'm confused here. For whatever reason I'm thinking that these two would equal the same thing if used with the same variables?

This is the part I don't understand. What "two" are you talking about? 5x² and cxⁿ? Yes, if c= 5 and n= 2 in the latter, they are the same. And, of course, 10x is the same as 5(2x^2-1). As far as it being "a special case of a constant times a function", it surely is: cxⁿ is the constant, c, times the function xⁿ.

Could somebody tell me what my book means when it says this?