# Differentiating cotx: Help Solving Wrong Answer

• wolfspirit
In summary, the conversation discusses how to differentiate cot(x) and the incorrect steps taken to do so. The correct answer is -csc^2(x). The mistake is using the chain rule instead of the quotient rule.

#### wolfspirit

I keep getting the wrong answer when i try to differentiate cotx..
this is what i get:
cotx = 1/tanx =cosx/sinx=cosx ⋅ sin^-1
so by the product and chain rule we have:
sinx⋅(sin x)^-1+cos⋅(-1sin^2 x)^-1 ⋅(cosx)^-1

=

sinx/sinx - cosx/cosx ⋅ sin^2x
=1-1/sin^2 x

where as the correct answer is -1/sin^2x = -csc^2 x

could someone please tell me where i am going wrong?

many thanks
Ryan

• Mark44
wolfspirit said:
I keep getting the wrong answer when i try to differentiate cotx..
this is what i get:
cotx = 1/tanx =cosx/sinx=cosx ⋅ sin^-1
so by the product and chain rule we have:
sinx⋅(sin x)^-1+cos⋅(-1sin^2 x)^-1 ⋅(cosx)^-1
d/dx(cos(x)) = -sin(x). It looks like you have other mistakes as well. For this problem it's probably simpler to use the quotient rule. You don't need to use the chain rule when you do so.
wolfspirit said:
=

sinx/sinx - cosx/cosx ⋅ sin^2x
=1-1/sin^2 x

where as the correct answer is -1/sin^2x = -csc^2 x

could someone please tell me where i am going wrong?

many thanks
Ryan

wolfspirit said:
I keep getting the wrong answer when i try to differentiate cotx..
this is what i get:
cotx = 1/tanx =cosx/sinx=cosx ⋅ sin^-1
so by the product and chain rule we have:
sinx⋅(sin x)^-1+cos⋅(-1sin^2 x)^-1 ⋅(cosx)^-1

=

sinx/sinx - cosx/cosx ⋅ sin^2x
=1-1/sin^2 x

where as the correct answer is -1/sin^2x = -csc^2 x

could someone please tell me where i am going wrong?

many thanks
Ryan

You have cot(x) = cos(x) * sin-1(x) = u * v

u = cos (x)
v = sin-1(x)

u' = -sin(x)
v' = -sin-2(x) * cos (x) [from the chain rule]

d(cot(x))/dx = u * v' + v * u' = -cos2(x)*sin-2(x) - sin(x) * sin-1(x) = -cot2(x) - 1 = -[1 + cot2(x)]

cot2(x) = cos2(x) / sin2(x)

1 + cot2(x) = 1 + cos2(x)/sin2(x) = [sin2(x) + cos2(x)] / sin2(x) = 1/sin2(x) = csc2(x)

-[1 + cot2(x)] = -csc2(x) = d(cot(x))/dx

Q.E.D.