Is my chain rule proof that d/dx(cot x) = −csc² x correct?

[dylanhouse]

Homework Statement

Using the chain rule, prove that d/dx(cotx)= -csc^2x

Homework Equations

Chain rule

The Attempt at a Solution

Is this correct?

f(x)=cotx=(tanx)^(-1)
Let f(x) = (x)^-1 Therefore, f'(x)= -1/(x^2)
Let g(x) = tanx Therefore, g'(x)= sec^2x

F'(x)=f'(g(x))g'(x)
=(-1/(tanx)^2)(secx^2)
=(-secx^2)/(tanx^2)
= -csc^2 x

 

[HallsofIvy]

Homework Statement

Using the chain rule, prove that d/dx(cotx)= -csc^2x

Homework Equations

Chain rule

The Attempt at a Solution

Is this correct?

f(x)=cotx=(tanx)^(-1)
Let f(x) = (x)^-1 Therefore, f'(x)= -1/(x^2)

It's not a good idea to use "x" as the variable here. Use, say, u instead:
f(u)= u^-1 so f'= -1u^-2.

Let g(x) = tanx Therefore, g'(x)= sec^2x

F'(x)=f'(g(x))g'(x)
=(-1/(tanx)^2)(secx^2)
=(-secx^2)/(tanx^2)
= -csc^2 x

Yes, that is a valid proof.