d/dx(cotx)
csc²x
We can find the derivative of cot(x) using the quotient rule. Recall that cot(x) = cos(x)/sin(x). Applying the quotient rule, we have:
d/dx(cot(x)) = [sin(x)d/dx(cos(x)) – cos(x)d/dx(sin(x))]/[sin^2(x)]
Next, we need to find the derivatives of cos(x) and sin(x):
d/dx(cos(x)) = -sin(x)
d/dx(sin(x)) = cos(x)
Substituting these into the formula for d/dx(cot(x)), we have:
d/dx(cot(x)) = [sin(x)(-sin(x)) – cos(x)(cos(x))]/[sin^2(x)]
Simplifying, we get:
d/dx(cot(x)) = -[cos^2(x) + sin^2(x)]/[sin^2(x)]
Recall the trig identity that cos^2(x) + sin^2(x) = 1, we can simplify the derivative further:
d/dx(cot(x)) = -1/[sin^2(x)]
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