d/dx [cot x]
To find the derivative of the function f(x) = cot(x), we can use the rules of differentiation
To find the derivative of the function f(x) = cot(x), we can use the rules of differentiation. However, since cot(x) can be rewritten as cos(x)/sin(x), we should start by finding the derivative of cos(x) and sin(x) separately.
1. Derivative of cos(x):
Using the chain rule, we differentiate the function g(x) = cos(x) as follows:
d/dx[cos(x)] = -sin(x)
Note that the derivative of cos(x) is -sin(x).
2. Derivative of sin(x):
Using the chain rule, we differentiate the function h(x) = sin(x) as follows:
d/dx[sin(x)] = cos(x)
Note that the derivative of sin(x) is cos(x).
Now, let’s find the derivative of f(x) = cot(x):
f(x) = cot(x) = cos(x)/sin(x)
Using the quotient rule, which states that the derivative of a quotient of two functions u(x) and v(x) is given by:
d/dx[u(x)/v(x)] = (u'(x)v(x) – u(x)v'(x)) / [v(x)]^2
Let’s differentiate f(x) using the quotient rule:
u(x) = cos(x) and v(x) = sin(x)
u'(x) = -sin(x) (from step 1)
v'(x) = cos(x) (from step 2)
Now, substitute these values into the quotient rule formula:
d/dx[cot(x)] = [(u'(x)v(x) – u(x)v'(x)) / [v(x)]^2
= [(-sin(x))(sin(x)) – (cos(x))(cos(x))] / [sin(x)]^2
Simplifying further:
d/dx[cot(x)] = [-sin^2(x) – cos^2(x)] / sin^2(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we have:
d/dx[cot(x)] = [-1] / sin^2(x)
Therefore, the derivative of f(x) = cot(x) is:
d/dx[cot(x)] = -1 / sin^2(x)
Note that this is a trigonometric identity, and we could have also obtained this result by rewriting f(x) in terms of sine and cosine functions.
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