Derivative of Cot(x) with Respect to x | Understanding and Applying the Quotient Rule in Calculus

d/dx cot x

To find the derivative of the function cot(x) with respect to x, we can use basic calculus rules and techniques

To find the derivative of the function cot(x) with respect to x, we can use basic calculus rules and techniques. Let’s start with the definition of the cotangent function:

cot(x) = 1 / tan(x)

Now, to differentiate cot(x), we will first express cot(x) in terms of sine and cosine:

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

Now, applying the quotient rule, which states that if we have a function u(x) = f(x) / g(x), then the derivative of u(x) is given by the formula:

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

Let’s apply this rule to differentiate cot(x):

f(x) = cos(x)
g(x) = sin(x)

f'(x) = -sin(x) (by differentiating cosine)
g'(x) = cos(x) (by differentiating sine)

Now, we can substitute these values into the derivative 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 rewrite the numerator:

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

Finally, we can use the reciprocal identity sin^2(x) = 1 / csc^2(x) to simplify the expression further:

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

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

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