How to Find the Derivative of the Cotangent Function Using the Quotient Rule and Trigonometric Identities

derivative of cotangent

The derivative of the cotangent function can be determined using the quotient rule

The derivative of the cotangent function can be determined using the quotient rule. Let’s start by defining the cotangent function:

The cotangent function, written as cot(x), is equal to the cosine of x divided by the sine of x. So, cot(x) = cos(x) / sin(x).

Now, let’s find the derivative. We need to apply the quotient rule, which states that if we have a function u(x) divided by v(x), where u(x) and v(x) are both differentiable functions, then the derivative of this quotient is given by:

d/dx(u(x)/v(x)) = [v(x)*u'(x) – u(x)*v'(x)] / [v(x)]^2

Applying this to cot(x) = cos(x) / sin(x), we have:
u(x) = cos(x) and v(x) = sin(x)

Now, let’s find the derivatives of u(x) and v(x):

u'(x) = -sin(x) (derivative of cos(x))
v'(x) = cos(x) (derivative of sin(x))

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

d/dx(cot(x)) = [sin(x)*(-sin(x)) – cos(x)*cos(x)] / [sin(x)]^2

Simplifying further, we have:

d/dx(cot(x)) = [-sin^2(x) – cos^2(x)] / sin^2(x)

Recall the trigonometric identity sin^2(x) + cos^2(x) = 1, so we can rewrite this as:

d/dx(cot(x)) = -1 / sin^2(x)

Alternatively, we can express the derivative of cot(x) using the reciprocal property of the tangent function, where cot(x) = 1 / tan(x). In this case, the derivative is:

d/dx(cot(x)) = d/dx(1 / tan(x)) = -sec^2(x)

So, the derivative of the cotangent function can be expressed in terms of sin(x) and cos(x) as -1 / sin^2(x), or in terms of the secant function as -sec^2(x).

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