d/dx cotx
To find the derivative of the cotangent function, we can start by expressing it in terms of sine and cosine functions
To find the derivative of the cotangent function, we can start by expressing it in terms of sine and cosine functions.
The cotangent function (cot(x)) is equal to 1 divided by the tangent function (tan(x)). The tangent function can be written as sine (sin(x)) divided by cosine (cos(x)). Therefore, cot(x) = cos(x) / sin(x).
To differentiate the cotangent function, we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), 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
Applying this rule to cot(x) = cos(x) / sin(x), we have:
cot'(x) = (cos'(x) * sin(x) – cos(x) * sin'(x)) / (sin(x))^2
Now, let’s calculate the derivatives of cosine and sine functions:
cos'(x) = -sin(x) (derivative of cosine)
sin'(x) = cos(x) (derivative of sine)
Substituting these derivatives back into the cot'(x) expression, we get:
cot'(x) = (-sin(x) * sin(x) – cos(x) * cos(x)) / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / (sin(x))^2
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the expression further:
cot'(x) = (-1) / (sin(x))^2
= -1 / sin^2(x)
Therefore, the derivative of cot(x) with respect to x is -1 divided by the square of the sine function, which can be written as -cosec^2(x) or -csc^2(x), where cosec(x) or csc(x) represents the reciprocal of the sine function.
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