Is this proof correct?

  • Thread starter dylanhouse
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  • #1
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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
 

Answers and Replies

  • #2
HallsofIvy
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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.
 

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