d csc(x)
The function csc(x) represents the cosecant of x, which is the reciprocal of the sine of x
The function csc(x) represents the cosecant of x, which is the reciprocal of the sine of x. In other words, csc(x) = 1/sin(x).
To find the derivative of csc(x) with respect to x, we can use the quotient rule.
The quotient rule states that if we have a function u(x) = f(x)/g(x), then the derivative of u(x) is given by:
u'(x) = (f'(x)g(x) – f(x)g'(x)) / (g(x))^2
Applying the quotient rule to csc(x), we have:
csc'(x) = (d/dx[1] * sin(x) – 1 * d/dx[sin(x)]) / (sin(x))^2
Here, d/dx[1] = 0, since 1 is a constant. The derivative of sin(x) with respect to x is cos(x), so we have:
csc'(x) = (0 * sin(x) – 1 * cos(x)) / (sin(x))^2
= -cos(x) / (sin(x))^2
Therefore, the derivative of csc(x) with respect to x is -cos(x) / (sin(x))^2.
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