Simplifying the Expression d cot(x) and Finding its Derivative: Definition, Rules, and Application

d cot(x)

To simplify the expression d cot(x), let’s begin by recalling the definition of the cotangent function

To simplify the expression d cot(x), let’s begin by recalling the definition of the cotangent function.

The cotangent function, cot(x), is defined as the reciprocal of the tangent function, tan(x). So, we can rewrite cot(x) in terms of tan(x) as:

cot(x) = 1/tan(x)

Now, let’s apply the derivative (d/dx) to both sides of the equation:

d/dx (cot(x)) = d/dx (1/tan(x))

To find the derivative of 1/tan(x), we can use the quotient rule. The quotient rule states that for functions f(x) = u(x)/v(x), the derivative is given by:

f'(x) = (u'(x) * v(x) – u(x) * v'(x)) / (v(x))^2

Applying this rule, let u(x) = 1 and v(x) = tan(x). We need to find u'(x) and v'(x) as well:

u'(x) = 0 (since the derivative of a constant is zero)
v'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))

Now, substitute the values into the quotient rule formula:

d/dx (cot(x)) = (0 * tan(x) – 1 * sec^2(x)) / (tan(x))^2

Simplifying:

d/dx (cot(x)) = – sec^2(x) / tan^2(x)

To manipulate the expression further, we can use the Pythagorean identity: tan^2(x) + 1 = sec^2(x). Rearranging it, we get sec^2(x) = tan^2(x) + 1.

Substitute sec^2(x) in terms of tan^2(x) in our derivative expression:

d/dx (cot(x)) = – (tan^2(x) + 1) / tan^2(x)

Expanding the denominator:

d/dx (cot(x)) = – (tan^2(x) / tan^2(x) + 1/tan^2(x))

Simplifying further:

d/dx (cot(x)) = – (1 + cot^2(x)) / cot^2(x)

Hence, the derivative of cot(x) with respect to x is given by – (1 + cot^2(x)) / cot^2(x).

It’s important to note that the domain of the cotangent function is restricted to values where tan(x) is not equal to zero. This is because cot(x) = 1/tan(x), and division by zero is undefined.

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