Derivative of:cot(x)
To find the derivative of the function cot(x), we need to use the basic differentiation rules
To find the derivative of the function cot(x), we need to use the basic differentiation rules. The cotangent function can also be expressed as the reciprocal of the tangent function: cot(x) = 1/tan(x).
Let’s start by differentiating 1/tan(x). To do this, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative 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). Therefore, g'(x) = 0 (since the derivative of a constant is always zero) and h'(x) = sec^2(x) (since the derivative of tan(x) is sec^2(x)). Substituting these values into the quotient rule, we get:
cot'(x) = (0 * tan(x) – 1 * sec^2(x)) / (tan(x))^2
= -sec^2(x) / tan^2(x)
Now, we need to simplify this expression using trigonometric identities. One such identity is: sec^2(x) = 1 + tan^2(x). Substituting this into our expression, we get:
cot'(x) = – (1 + tan^2(x)) / tan^2(x)
= -1/tan^2(x) – 1
Finally, we can rewrite this in terms of the cotangent function:
cot'(x) = -cot^2(x) – 1
Therefore, the derivative of cot(x) is -cot^2(x) – 1.
More Answers:
[next_post_link]