The Derivative of Cot(x): Using the Quotient Rule and Trigonometric Identities

derivative of cot(x)

To find the derivative of cot(x), we will make use of the quotient rule and the derivative of sine and cosine

To find the derivative of cot(x), we will make use of the quotient rule and the derivative of sine and cosine.

The cotangent function can be expressed as the ratio of cosine and sine:

cot(x) = cos(x) / sin(x)

Using the quotient rule, the derivative of cot(x) can be found as:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

where g(x) = cos(x) and h(x) = sin(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = -sin(x) (derivative of cosine is -sin)
h'(x) = cos(x) (derivative of sine is cos)

Substituting these derivatives into the quotient rule formula, we get:

f'(x) = (-sin(x) * sin(x) – cos(x) * cos(x)) / (sin(x))^2

Next, we can simplify the numerator:

f'(x) = (-sin^2(x) – cos^2(x)) / sin^2(x)

Remembering the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator as:

f'(x) = -1 / sin^2(x)

Finally, since cosecant (csc) is the reciprocal of sine, we can write f'(x) as:

f'(x) = -1 / (1/sin^2(x))

f'(x) = -sin^2(x) / 1

f'(x) = -sin^2(x)

So, the derivative of cot(x) is -sin^2(x).

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