A Guide to Finding the Derivative of Cot(x) Using the Quotient Rule

d/dx (cot x)

To find the derivative of cot(x), we can use the quotient rule

To find the derivative of cot(x), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:

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

To apply this rule to cot(x), we need to express cot(x) as a quotient of two functions. The reciprocal identity for tangent tells us that cot(x) is equal to 1/tan(x), so we can write cot(x) as:

cot(x) = 1/tan(x)

Now, we can differentiate cot(x) by applying the quotient rule. Let g(x) = 1 and h(x) = tan(x). The derivatives are:

g'(x) = 0 (since g(x) is a constant)
h'(x) = sec^2(x) (by the derivative of tangent rule)

Now, plugging these values into the quotient rule formula, we have:

cot'(x) = (0*tan(x) – 1*sec^2(x))/(tan(x))^2
= -sec^2(x)/(tan(x))^2

To simplify further, we can use the Pythagorean identity for trigonometric functions: 1 + tan^2(x) = sec^2(x). Rearranging this equation, we get sec^2(x) = 1 + tan^2(x).

Substituting this into our derivative, we have:

cot'(x) = -1/(1 + tan^2(x))

So, the derivative of cot(x) is -1/(1 + tan^2(x)).

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