Understanding the Derivative of cot(x) and How to Calculate it Using the Quotient Rule.

Derivitive of 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 you have a function of the form f(x) = g(x) / h(x), the derivative of f(x) is given by:

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

Let’s apply the quotient rule to find the derivative of cot(x):

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

Using the quotient rule, we can find the derivative of cot(x) as:

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

We know that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). Substituting these values, we get:

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

We can simplify this expression further:

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

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

f'(x) = – [1] / [sin^2(x)]

Finally, using the reciprocal identity sin^2(x) = 1 / csc^2(x), we can rewrite the derivative as:

f'(x) = – [1] / [1 / csc^2(x)]

Simplifying further, we get:

f'(x) = – csc^2(x)

Thus, the derivative of cot(x) is – csc^2(x).

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