Derivative of cot(x) Explained: Using the Quotient Rule of Differentiation

d/dx cotx

To find the derivative of the function cot(x) with respect to x, we can use the quotient rule of differentiation

To find the derivative of the function cot(x) with respect to x, we can use the quotient rule of differentiation.

The quotient rule states that if you have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

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

In our case, g(x) = 1 and h(x) = tan(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = 0 (since g(x) = 1 is a constant)

h'(x) = sec^2(x) (using the derivative of tan(x), which is sec^2(x))

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

f'(x) = (0 * tan(x) – 1 * sec^2(x)) / [tan(x)]^2

Since 0 * tan(x) is always 0, we can simplify this expression:

f'(x) = – sec^2(x) / [tan(x)]^2

Using the trigonometric identity for sec^2(x), which is 1 + tan^2(x), we can simplify further:

f'(x) = – (1 + tan^2(x)) / [tan(x)]^2

Now, we can use the identity that cot(x) = 1/tan(x) to rewrite the expression:

f'(x) = – (1 + cot^2(x))

So, the derivative of cot(x) with respect to x is – (1 + cot^2(x)).

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