d/dx cot(x)
To differentiate the function cot(x) with respect to x, we can use the quotient rule
To differentiate the function cot(x) with respect to x, we can use the quotient rule.
The quotient rule states that for a function f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, 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 differentiate cot(x) using the quotient rule. In this case, g(x) = 1 and h(x) = tan(x), since cot(x) = 1/tan(x).
Using the quotient rule, we have:
cot'(x) = [1*tan(x) – 1*(-sec^2(x))] / [tan(x)]^2
To simplify this further, we can rewrite sec^2(x) as 1 + tan^2(x) using the identity sec^2(x) = 1 + tan^2(x).
cot'(x) = [tan(x) + sec^2(x)] / [tan(x)]^2
Since sec^2(x) = 1 + tan^2(x), we can substitute this back into the expression:
cot'(x) = [tan(x) + 1 + tan^2(x)] / [tan(x)]^2
Simplifying and combining like terms, we get:
cot'(x) = [1 + tan^2(x)] / [tan(x)]^2
Using the Pythagorean identity tan^2(x) + 1 = sec^2(x), we can simplify further:
cot'(x) = [sec^2(x)] / [tan(x)]^2
Finally, we can rewrite sec^2(x) as 1 + tan^2(x) one more time:
cot'(x) = [1 + tan^2(x)] / [tan(x)]^2
This is the derivative of cot(x) with respect to x.
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