deriv of cscx
cscxcotx
The derivative of csc(x) is:
d/dx[csc(x)] = -csc(x)cot(x)
To derive this formula, we start by expressing csc(x) in terms of sine and cosine functions. Recall that csc(x) is defined as the reciprocal of sin(x):
csc(x) = 1/sin(x)
From here, we use the quotient rule to differentiate csc(x) with respect to x:
d/dx[csc(x)] = d/dx[1/sin(x)] = -1/sin^2(x) * d/dx[sin(x)]
Now, we can use the chain rule to differentiate sin(x):
d/dx[csc(x)] = -1/sin^2(x) * cos(x)
Finally, we use the identity cot(x) = cos(x)/sin(x) to simplify the expression:
d/dx[csc(x)] = -1/sin(x) * cos(x)/sin(x) = -csc(x)cot(x)
Therefore, the derivative of csc(x) is -csc(x)cot(x).
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