d/dx[cotx]
To find the derivative of cot(x) with respect to x, we can use the quotient rule
To find the derivative of cot(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x))/(h(x))^2
In this case, g(x) = 1 and h(x) = tan(x). Therefore, g'(x) = 0 and h'(x) = sec^2(x).
Plugging these values into the derivative formula, we get:
cot'(x) = (0 * tan(x) – 1 * sec^2(x))/(tan(x))^2
Simplifying further, we have:
cot'(x) = -sec^2(x)/(tan(x))^2
Now, we can rewrite sec^2(x) and tan(x) in terms of sin(x) and cos(x) using their definitions:
cot'(x) = -1/(sin^2(x)/cos^2(x))
Invert the fraction and simplify:
cot'(x) = -cos^2(x)/sin^2(x)
Finally, we can rewrite cos^2(x) and sin^2(x) using the Pythagorean identity:
cot'(x) = -(1 – sin^2(x))/sin^2(x)
Further simplifying:
cot'(x) = -1/sin^2(x) + 1
Therefore, the derivative of cot(x) with respect to x is -csc^2(x) + 1.
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