Welcome to a proof that the derivative
of cotangent x with respect to x
equals negative cosecant squared x.
First, rewrite cotangent x
using the quotient identity,
cotangent x equals cosine x
divided by sine x
and then we'll find the derivative
of cosine x divided by sine x
with respect to x using the quotient rule
which is stated here below for reference.
So the derivative of
cosine x divided sine x
with respect to x
is equal to the denominator
which is sine x
times the derivative of the numerator
which would be the derivative
of cosine x minus the numerator
which is cosine x
times the derivative of the denominator
which would be the derivative of sine x
all divided by the denominator squared
which would be sine x squared.
Now, we'll find the derivatives.
The derivative of cosine x
is equal to negative sine x
and the derivative of sine x
is equal to cosine x.
Notice how we also can
rewrite sine x squared
as sine squared x.
And now, we'll simplify the numerator.
Here we'll have negative sine squared x
and then, minus cosine squared x.
From here, we'll factor a negative,
or a negative one, out of the numerator.
Once we do this, we have negative,
or the opposite of, the quantity
sine squared x plus cosine squared x
all divided by sine squared x
and sine squared x plus cosine squared x
is equal to one
from a Pythagorean identity.
So the numerator
simplifies to negative one.
So we have negative one
divided sine squared x
which we can expand and write as
negative one over sine x
times one over sine x,
because remember one over sine x
is equal to cosecant x,
which gives us negative
cosecant squared x.
So we have our proof.
The derivative of cotangent
x with respect to x
equals negative cosecant squared x.
