Derivative of cot x
To find the derivative of the cotangent function with respect to x, we can use the quotient rule
To find the derivative of the cotangent function with respect to x, we can use the quotient rule. First, let’s recall the definitions of the cotangent function and the quotient rule:
1. Cotangent function:
The cotangent function of an angle is defined as the ratio of the cosine of the angle to the sine of the angle. Mathematically, the cotangent of an angle x is given by cot(x) = cos(x)/sin(x).
2. Quotient rule:
The quotient rule is a formula that allows us to find the derivative of a function that is a ratio of two other functions. If we 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.
Now, let’s apply the quotient rule to find the derivative of the cotangent function.
Derivative of cot(x):
Using the quotient rule with g(x) = cos(x) and h(x) = sin(x), we have:
cot'(x) = (cos'(x) * sin(x) – cos(x) * sin'(x)) / (sin(x))^2.
To simplify this, we need to find the derivatives of cos(x) and sin(x):
– The derivative of cos(x) is given by cos'(x) = -sin(x).
– The derivative of sin(x) is given by sin'(x) = cos(x).
Substituting these values into the quotient rule equation, we get:
cot'(x) = (-sin(x) * sin(x) – cos(x) * cos(x)) / (sin(x))^2.
Further simplifying the numerator, we have:
cot'(x) = (-sin^2(x) – cos^2(x)) / (sin(x))^2.
Now, recalling the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator as -1:
cot'(x) = -1 / (sin(x))^2.
Therefore, the derivative of the cotangent function is cot'(x) = -1 / (sin(x))^2.
This derivative expression represents the instantaneous rate of change of the cotangent function at any given angle x.
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