d/dx (cot(x) / sin(x)) =
I got dy/dx= -csc(x) - cos(x)cot(x)
I meant
dy/dx= -csc(x) - (cos(x))^2cot(x)
((-csc^2x)(sinx)-(cotx)(cosx))/sin^2x
(-1/sinx - cos^2x/sinx)/sin^2x
-(1+cos^2x)/sin^3x
-(cscx+cosx*cotx)/sin^2x
You seem to have lost that sin^2x in the denominator
To find the derivative of the given function, we can use the quotient rule.
Recall that the quotient rule states:
If we have a function f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x), denoted as f'(x), is given by:
f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
Now, let's apply the quotient rule to find the derivative of the given function:
Let g(x) = cot(x) and h(x) = sin(x)
g'(x) = -csc^2(x) (derivative of cot(x) is -csc^2(x), where csc(x) is the cosecant function)
h'(x) = cos(x) (derivative of sin(x) is cos(x))
Plugging these values into the quotient rule formula:
f'(x) = (sin(x) * (-csc^2(x)) - cot(x) * cos(x)) / (sin(x))^2
Simplifying:
f'(x) = (-sin(x) * csc^2(x) - cot(x) * cos(x)) / sin^2(x)
Now, let's simplify further using trigonometric identities:
Recall that csc(x) = 1/sin(x), so csc^2(x) = (1/sin(x))^2 = 1/sin^2(x)
Plugging this in:
f'(x) = (-sin(x) * (1/sin^2(x)) - cot(x) * cos(x)) / sin^2(x)
Simplifying:
f'(x) = (-1/sin(x) - cot(x) * cos(x)) / sin^2(x)
Recall that cot(x) = cos(x) / sin(x), so we can substitute cot(x) in terms of cos(x) and sin(x):
f'(x) = (-1/sin(x) - (cos(x) / sin(x)) * cos(x)) / sin^2(x)
Simplifying:
f'(x) = (-1/sin(x) - cos^2(x) / sin(x)) / sin^2(x)
At this point, we can combine the terms in the numerator:
f'(x) = (-1 - cos^2(x)) / sin(x) / sin^2(x)
Now, let's simplify the numerator by using the trigonometric identity sin^2(x) + cos^2(x) = 1:
f'(x) = (-1 - (1 - sin^2(x))) / sin(x) / sin^2(x)
Simplifying further:
f'(x) = (-1 - 1 + sin^2(x)) / sin(x) / sin^2(x)
f'(x) = (sin^2(x) - 2) / sin(x) / sin^2(x)
Finally, remember that sin^2(x) - 2 = -cos(2x) (double angle formula for cosine):
f'(x) = (-cos(2x)) / sin(x) / sin^2(x)
Therefore, the derivative of cot(x) / sin(x) is -cos(2x) / sin(x) / sin^2(x), which can also be written as -csc(x) - cos(x) * cot(x).