How to Find the Derivative of cot(x) using the Quotient Rule?

Derivative of cotx

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 two functions u(x) and v(x), then the derivative of their quotient is given by:

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

In this case, u(x) = 1 and v(x) = tan(x).

So, let’s start by finding the derivatives of u(x) and v(x):

u'(x) = 0, since the derivative of a constant is zero.

v'(x) = sec^2(x), which is the derivative of tan(x).

Now, let’s substitute these values into the quotient rule formula:

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

Simplifying this expression:

d/dx(cot(x)) = -sec^2(x) / [tan(x)]^2.

Recall that sec(x) is equal to 1 / cos(x) and tan(x) is equal to sin(x) / cos(x). Substituting these values, we get:

d/dx(cot(x)) = -[1 / cos^2(x)] / [(sin(x) / cos(x))^2].

Simplifying further:

d/dx(cot(x)) = -[1 / cos^2(x)] / [sin^2(x) / cos^2(x)].

Dividing by a fraction is equivalent to multiplying by its reciprocal:

d/dx(cot(x)) = -[1 / cos^2(x)] * [cos^2(x) / sin^2(x)].

The cos^2(x) terms cancel out:

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

Since sin^2(x) = 1 – cos^2(x), we can rewrite the expression as:

d/dx(cot(x)) = -1 / (1 – cos^2(x)).

Finally, using the trigonometric identity 1 – cos^2(x) = sin^2(x), the derivative of cot(x) can be simplified to:

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

Therefore, the derivative of cot(x) is -csc^2(x).

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