Discovering the Derivative of Cot(x) Using the Quotient Rule in Trigonometry

d/dx cot(x)

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

To find the derivative of cot(x), we will use the quotient rule. The quotient rule states that if we have the function f(x) = g(x)/h(x), then its derivative can be found using the formula:

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

Let’s apply this rule to cot(x):

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

Using the quotient rule, we need to find the derivatives of cos(x) and sin(x):

1. The derivative of cos(x) is -sin(x).

2. The derivative of sin(x) is cos(x).

Now, substituting these derivatives into the quotient rule formula:

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

Simplifying further:

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

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can simplify:

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

Finally, we can express the derivative of cot(x) in terms of cot(x) itself:

cot'(x) = -1 / (sin^2(x)) = -csc^2(x)

Therefore, the derivative of cot(x) is equal to -csc^2(x), where csc(x) represents the cosecant function.

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